Explanation:
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, let me recall how to do that. I think polar coordinates are given by (r, θ), where r is the distance from the origin to the point, and θ is the angle made with the positive x-axis.

First, I remember there are formulas to convert between rectangular (x, y) and polar (r, θ). Let me try to remember them. I think r is calculated using the Pythagorean theorem, right? So r equals the square root of x squared plus y squared. Then θ is the arctangent of y over x. But wait, I need to be careful with the arctangent because it depends on the quadrant where the point is located.

So let me start with the given point (0, 3). Let me note that x is 0 and y is 3. Let me plug these into the formula for r. So r = sqrt(x² + y²) = sqrt(0² + 3²) = sqrt(0 + 9) = sqrt(9) = 3. Okay, so r is 3. That seems straightforward.

Now for θ. The formula is θ = arctan(y/x). But here, x is 0. Wait, arctan(y/x) would be arctan(3/0). But division by zero is undefined. Hmm, so how do I handle that?

I remember that when x is 0, the point lies on the y-axis. So if x is 0 and y is positive, the angle should be π/2 radians, right? Because the positive y-axis is π/2 radians from the positive x-axis. Similarly, if y were negative, it would be 3π/2. But in this case, y is 3, so θ should be π/2.

But let me verify that. Let me think about the unit circle. The point (0, 3) is straight up along the y-axis. So if I imagine the angle starting from the positive x-axis, rotating counterclockwise to the positive y-axis, that's 90 degrees, which is π/2 radians. So θ is π/2.

But just to make sure I didn't make a mistake with the arctangent formula. Since x is 0, the arctangent of y/x is undefined, but we can consider the limit as x approaches 0 from the positive side. If x approaches 0 from the positive side, then arctan(y/x) approaches arctan(infinity) which is π/2. Similarly, if x approaches 0 from the negative side, arctan(y/x) approaches arctan(-infinity), which is -π/2, but since we are in the positive y-axis, we need to adjust the angle accordingly.

But since the original point is (0, 3), which is in the positive y-axis, so we need to make sure θ is in the correct quadrant. Since x is 0 and y is positive, the point is on the positive y-axis, so θ is π/2.

Alternatively, maybe I can use the arctangent function with two arguments, which takes into account the quadrant. But since I don't have a calculator here, I need to think.

Alternatively, maybe I can use the fact that tan θ = y/x. But since x is 0, tan θ is undefined, which occurs at θ = π/2 + kπ. But since the point is in the positive y-axis, θ is π/2.

So, putting it all together, r is 3 and θ is π/2. Therefore, the polar coordinates should be (3, π/2).

Wait, let me check if there's another possible angle. Since angles are periodic with 2π, but θ is supposed to be between 0 and 2π. π/2 is within that range. Also, since r must be positive, 3 is positive. So that's good.

But let me think again. If I use the formula θ = arctan(y/x), but since x is 0, maybe I need to use another approach. For example, if x is 0, then θ is π/2 if y is positive, and 3π/2 if y is negative. So since y is positive here, θ is π/2.

Alternatively, if I use the formula for θ, which is arctan(y/x) adjusted by the quadrant. Since x is 0 and y is positive, the point is on the positive y-axis, so θ is π/2.

Therefore, I think my answer is correct.

But just to make sure, let me visualize the point (0, 3). In the rectangular coordinate system, that's 3 units up along the y-axis. In polar coordinates, that would be a radius of 3 units from the origin, making a 90-degree angle with the positive x-axis. So yes, that's π/2 radians.

Another way to check is to convert back from polar to rectangular coordinates. If I have (r, θ) = (3, π/2), then x = r cos θ = 3 cos(π/2) = 3 * 0 = 0, and y = r sin θ = 3 sin(π/2) = 3 * 1 = 3. So that gives me back (0, 3), which matches the original point.

Therefore, my conversion seems correct.

I don't think I made any mistakes here. The key was recognizing that when x is 0, the angle is either π/2 or 3π/2 depending on the sign of y. Since y is positive here, it's π/2.

So the final answer should be (3, π/2).

**Final Answer**
The polar coordinates are \boxed{(3, \frac{\pi}{2})}.

To convert the rectangular coordinates $(0, 3)$ to polar coordinates $(r, \theta)$, we use the standard formulas for conversion:

- $ r = \sqrt{x^2 + y^2} $
- $ \theta = \arctan\left(\frac{y}{x}\right) $, adjusted for the correct quadrant

---

### Step 1: Compute $ r $

Given the point $(x, y) = (0, 3)$, we calculate:

$$
r = \sqrt{0^2 + 3^2} = \sqrt{9} = 3
$$

So, the radial distance from the origin is $ r = 3 $.

---

### Step 2: Compute $ \theta $

We use the formula:

$$
\theta = \arctan\left(\frac{y}{x}\right)
$$

However, in this case, $ x = 0 $, so the expression becomes:

$$
\theta = \arctan\left(\frac{3}{0}\right)
$$

This is undefined, but we can interpret it geometrically. The point $(0, 3)$ lies on the **positive y-axis**, which corresponds to an angle of $ \frac{\pi}{2} $ radians from the positive x-axis.

To confirm, we can also consider the unit circle: the point $(0, 3)$ is directly above the origin on the y-axis, so the angle is indeed $ \frac{\pi}{2} $.

---

### Step 3: Final Polar Coordinates

We now have:

- $ r = 3 $
- $ \theta = \frac{\pi}{2} $

Since $ r > 0 $ and $ 0 \leq \theta < 2\pi $, the polar coordinates are valid.

---

### Final Answer

$$
\boxed{(3, \frac{\pi}{2})}
$$

[Assistant]
Explanation:
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, let me recall how to do that. I think polar coordinates are given by (r, θ), where r is the distance from the origin to the point, and θ is the angle made with the positive x-axis.

First, I remember there are formulas to convert between rectangular (x, y) and polar (r, θ). Let me try to remember them. I think r is calculated using the Pythagorean theorem, right? So r equals the square root of x squared plus y squared. Then θ is the arctangent of y over x. But wait, I need to be careful with the arctangent because it depends on the quadrant where the point is located.

So let me start with the given point (0, 3). Let me note that x is 0 and y is 3. Let me plug these into the formula for r. So r = sqrt(x² + y²) = sqrt(0² + 3²) = sqrt(0 + 9) = sqrt(9) = 3. Okay, so r is 3. That seems straightforward.

Now for θ. The formula is θ = arctan(y/x). But here, x is 0. Wait, arctan(y/x) would be arctan(3/0). But division by zero is undefined. Hmm, so how do I handle that?

I remember that when x is 0, the point lies on the y-axis. So if x is 0 and y is positive, the angle should be π/2 radians, right? Because the positive y-axis is π/2 radians from the positive x-axis. Similarly, if y were negative, it would be 3π/2. But in this case, y is 3, so θ should be π/2.

But let me verify that. Let me think about the unit circle. The point (0, 3) is straight up along the y-axis. So if I imagine the angle starting from the positive x-axis, rotating counterclockwise to the positive y-axis, that's 90 degrees, which is π/2 radians. So θ is π/2.

But just to make sure I didn't make a mistake with the arctangent formula. Since x is 0, the arctangent of y/x is undefined, but we can consider the limit as x approaches 0 from the positive side. If x approaches 0 from the positive side, then arctan(y/x) approaches arctan(infinity) which is π/2. Similarly, if x approaches 0 from the negative side, arctan(y/x) approaches arctan(-infinity), which is -π/2, but since we are in the positive y-axis, we need to adjust the angle accordingly.

But since the original point is (0, 3), which is in the positive y-axis, so we need to make sure θ is in the correct quadrant. Since x is 0 and y is positive, the point is on the positive y-axis, so θ is π/2.

Alternatively, maybe I can use the arctangent function with two arguments, which takes into account the quadrant. But since I don't have a calculator here, I need to think.

Alternatively, maybe I can use the fact that tan θ = y/x. But since x is 0, tan θ is undefined, which occurs at θ = π/2 + kπ. But since the point is in the positive y-axis, θ is π/2.

So, putting it all together, r is 3 and θ is π/2. Therefore, the polar coordinates should be (3, π/2).

Wait, let me check if there's another possible angle. Since angles are periodic with 2π, but θ is supposed to be between 0 and 2π. π/2 is within that range. Also, since r must be positive, 3 is positive. So that's good.

But let me think again. If I use the formula θ = arctan(y/x), but since x is 0, maybe I need to use another approach. For example, if x is 0, then θ is π/2 if y is positive, and 3π/2 if y is negative. So since y is positive here, θ is π/2.

Alternatively, if I use the formula for θ, which is arctan(y/x) adjusted by the quadrant. Since x is 0 and y is positive, the point is on the positive y-axis, so θ is π/2.

Therefore, I think my answer is correct.

But just to make sure, let me visualize the point (0, 3). In the rectangular coordinate system, that's 3 units up along the y-axis. In polar coordinates, that would be a radius of 3 units from the origin, making a 90-degree angle with the positive x-axis. So yes, that's π/2 radians.

Another way to check is to convert back from polar to rectangular coordinates. If I have (r, θ) = (3, π/2), then x = r cos θ = 3 cos(π/2) = 3 * 0 = 0, and y = r sin θ = 3 sin(π/2) = 3 * 1 = 3. So that gives me back (0, 3), which matches the original point.

Therefore, my conversion seems correct.

I don't think I made any mistakes here. The key was recognizing that when x is 0, the angle is either π/2 or 3π/2 depending on the sign of y. Since y is positive here, it's π/2.

So the final answer should be (3, π/2).

**Final Answer**
The polar coordinates are \boxed{(3, \frac{\pi}{2})}.

To convert the rectangular coordinates $(0, 3)$ to polar coordinates $(r, \theta)$, we use the standard formulas for conversion:

- $ r = \sqrt{x^2 + y^2} $
- $ \theta = \arctan\left(\frac{y}{x}\right) $, adjusted for the correct quadrant

---

### Step 1: Compute $ r $

Given the point $(x, y) = (0, 3)$, we calculate:

$$
r = \sqrt{0^2 + 3^2} = \sqrt{9} = 3
$$

So, the radial distance from the origin is $ r = 3 $.

---

### Step 2: Compute $ \theta $

We use the formula:

$$
\theta = \arctan\left(\frac{y}{x}\right)
$$

However, in this case, $ x = 0 $, so the expression becomes:

$$
\theta = \arctan\left(\frac{3}{0}\right)
$$

This is undefined, but we can interpret it geometrically. The point $(0, 3)$ lies on the **positive y-axis**, which corresponds to an angle of $ \frac{\pi}{2} $ radians from the positive x-axis.

To confirm, we can also consider the unit circle: the point $(0, 3)$ is directly above the origin on the y-axis, so the angle is indeed $ \frac{\pi}{2} $.

---

### Step 3: Final Polar Coordinates

We now have:

- $ r = 3 $
- $ \theta = \frac{\pi}{2} $

Since $ r > 0 $ and $ 0 \leq \theta < 2\pi $, the polar coordinates are valid.

---

### Final Answer

$$
\boxed{(3, \frac{\pi}{2})}
$$

Solution:
We have that $r = \sqrt{0^2 + 3^2} = 3.$  Also, if we draw the line connecting the origin and $(0,3),$ this line makes an angle of $\frac{\pi}{2}$ with the positive $x$-axis.

[asy]
unitsize(0.8 cm);

draw((-0.5,0)--(3.5,0));
draw((0,-0.5)--(0,3.5));
draw(arc((0,0),3,0,90),red,Arrow(6));

dot((0,3), red);
label("$(0,3)$", (0,3), W);
dot((3,0), red);
[/asy]

Therefore, the polar coordinates are $\boxed{\left( 3, \frac{\pi}{2} \right)}.$


Explanation:
Okay, so I need to figure out which student has the greatest average speed based on the graph provided. Let me start by recalling what average speed means. Average speed is calculated by dividing the total distance traveled by the total time taken. So, in this case, since the graph is a distance-time graph, the average speed for each student would be the slope of the line connecting their starting point to their ending point. Wait, but actually, since each student is represented by a single dot on the graph, does that mean they all started at the same time and ran until the same time? Or is each dot representing their position at a specific time?

Looking at the Asymptote code, there are dots at different coordinates. For example, Evelyn is at (1.25, 4.5), Briana at (2.5, 2.2), Carla at (4.25, 5.2), Debra at (5.6, 2.8), and Angela at (6.8, 1.4). The axes are labeled with time on the x-axis and distance on the y-axis. The graph has vertical lines from x=0 to x=8 and horizontal lines from y=0 to y=6. The labels for the axes are "distance" on the left (y-axis) and "time" on the bottom (x-axis).

So, each student's position is plotted at a specific time and distance. But wait, if they are just single points, how do we determine their average speed? Maybe the graph is showing the distance covered at different times, so each dot represents the distance covered by each student at a particular time. But if that's the case, then to find average speed, we need to know the total distance each student covered over the total time. However, the problem is that the graph doesn't show the starting point. Wait, but maybe all students started at time 0? If they all started at time 0, then their positions at time t would be their distance covered. But the problem is that the graph only shows one point for each student. So, unless they all finished at the same time, but the x-coordinate of each dot is different. For example, Evelyn is at x=1.25, Briana at x=2.5, Carla at x=4.25, Debra at x=5.6, and Angela at x=6.8. So, each student's dot is at a different time.

Wait, but if the graph is of a training run, maybe each student ran for a different amount of time, and the dot represents their position at the end of their run? If that's the case, then the average speed would be the distance divided by the time. So, for each student, their average speed is (distance)/time.

But then, how do we know the total distance? For example, Evelyn's dot is at (1.25, 4.5). So, that would mean at time 1.25 units, she had covered 4.5 units of distance. So, if we assume that she started at time 0, distance 0, then her average speed would be 4.5 / 1.25. Similarly for the others.

But wait, the problem says "the results of a cross-country team's training run are graphed below." So, maybe each student's run is represented by a single point? That seems odd. Usually, a distance-time graph would show the position over time, so if they are plotting a single point, maybe it's the final position? But then, if they all started at the same time, but different times? Wait, maybe the graph is showing the distance covered by each student at different times? But how can we know the total time each student ran?

Alternatively, maybe the graph is showing the distance covered by each student at the same time? But that would be a different interpretation.

Wait, let me think again. The Asymptote code draws a grid with vertical lines from x=0 to x=8 and horizontal lines from y=0 to y=6. The axes are labeled with time on the x-axis and distance on the y-axis. The dots are plotted at different coordinates. Each dot is labeled with the student's name. So, perhaps each dot represents the position of the student at a specific time. For example, Evelyn is at (1.25, 4.5), meaning that at time 1.25 units, she was at distance 4.5 units. Similarly, Briana is at (2.5, 2.2), so at time 2.5 units, she was at distance 2.2 units.

But if that's the case, then to find the average speed, we need to know the total distance each student covered over the total time they ran. However, the problem is that we don't know when each student started or when they stopped. If they all started at time 0, then their average speed would be their distance at the time they stopped divided by the time they stopped. But if they started at different times, we can't know.

Wait, but maybe the graph is showing the distance covered by each student at the end of their run. So, for example, Evelyn ran for 1.25 units of time and covered 4.5 units of distance. Then her average speed is 4.5 / 1.25. Similarly, Briana ran for 2.5 units of time and covered 2.2 units of distance, so her average speed is 2.2 / 2.5. Carla ran for 4.25 units of time and covered 5.2 units of distance, so 5.2 / 4.25. Debra ran for 5.6 units of time and covered 2.8 units of distance, so 2.8 / 5.6. Angela ran for 6.8 units of time and covered 1.4 units of distance, so 1.4 / 6.8.

If that's the case, then we can calculate each student's average speed and compare them.

Alternatively, maybe the graph is showing the distance covered at different times, but each student's run is represented by a single point. But that would not make sense for average speed unless we assume they started at time 0 and stopped at their respective times.

Alternatively, maybe the graph is showing the distance covered by each student at the same time? For example, at time t, each student's position is plotted. But then, if they all ran until the same time, but the graph only shows one point for each student, that would be their position at that time. But then, how do we know their total distance?

But given that the problem is asking for average speed, which is total distance over total time, and given that the graph is a distance-time graph, the most logical interpretation is that each student's dot represents their position at a specific time. If we assume that each student started at time 0, distance 0, then their average speed would be their distance divided by their time.

Alternatively, if they didn't start at time 0, but the graph is showing their position at some time, but without knowing their starting time, we can't calculate average speed. However, since the problem is presented in a way that expects an answer, it's likely that we are supposed to assume that each student started at time 0, and the dot represents their position at the end of their run. Therefore, their average speed is distance divided by time.

So, proceeding with that assumption, let's calculate each student's average speed.

First, let's note down each student's coordinates:

- Evelyn: (1.25, 4.5)
- Briana: (2.5, 2.2)
- Carla: (4.25, 5.2)
- Debra: (5.6, 2.8)
- Angela: (6.8, 1.4)

Assuming that the x-coordinate is time and the y-coordinate is distance, then:

For Evelyn: time = 1.25, distance = 4.5. Average speed = 4.5 / 1.25

Let me compute that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 1.25 times 3.6 is 4.5. So 4.5 / 1.25 = 3.6 units per time unit.

For Briana: time = 2.5, distance = 2.2. Average speed = 2.2 / 2.5. Let me calculate that. 2.2 divided by 2.5. Well, 2.5 goes into 2.2 0.88 times. So 0.88 units per time unit.

For Carla: time = 4.25, distance = 5.2. Average speed = 5.2 / 4.25. Let me compute that. 4.25 times 1.2 is 5.1, so 5.2 / 4.25 is approximately 1.2235 units per time unit.

For Debra: time = 5.6, distance = 2.8. Average speed = 2.8 / 5.6 = 0.5 units per time unit.

For Angela: time = 6.8, distance = 1.4. Average speed = 1.4 / 6.8. Let me calculate that. 1.4 divided by 6.8 is approximately 0.2059 units per time unit.

So, comparing all these:

Evelyn: 3.6

Briana: 0.88

Carla: ~1.22

Debra: 0.5

Angela: ~0.206

So, Evelyn has the highest average speed. Therefore, the answer should be Evelyn.

But wait, let me check if my interpretation is correct. Because if the graph is not showing the total distance covered by each student, but instead, maybe each student's position at a certain time, but they might have different starting times? For example, maybe some students started later? But the problem doesn't mention anything about that.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the question is about the average speed during the entire training run. However, without knowing the total time each student ran, we can't compute average speed.

But given that the problem is presented with a graph where each student is represented by a single point, and the axes are time and distance, the most logical assumption is that each student's run is represented by a single point, meaning that they all started at time 0, and the dot is their position at the end of their run. Therefore, their average speed is distance divided by time.

Alternatively, if the graph is showing the distance covered at different times, but each student's run is represented by a single point, maybe the graph is showing the distance covered at the same time? For example, at time t, each student's distance is plotted. But then, if they all ran until the same time, but the graph only shows one point for each student, then we would need to know their total distance. However, in that case, the average speed would be total distance divided by total time. But if the graph is showing their position at a specific time, then unless we know their starting point, we can't compute average speed.

But since the problem is from a cross-country team's training run, it's possible that each student ran for a different duration, and the graph is showing their final position. Therefore, assuming they all started at time 0, distance 0, then their average speed is distance divided by time.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the question is about the average speed during the entire run. However, without knowing the total time each student ran, we can't compute average speed.

But given that the problem is presented with the Asymptote code, which plots the points with specific coordinates, and given that the answer is expected, I think the first interpretation is correct.

Therefore, proceeding with that, Evelyn has the highest average speed of 3.6, so the answer is Evelyn.

But let me check again the calculations to make sure I didn't make any arithmetic errors.

Evelyn: 4.5 / 1.25. Let me do this division again. 1.25 times 3 is 3.75. 4.5 minus 3.75 is 0.75. 0.75 divided by 1.25 is 0.6. So total is 3.6. Correct.

Briana: 2.2 / 2.5. Multiply numerator and denominator by 2 to eliminate decimal: 4.4 / 5 = 0.88. Correct.

Carla: 5.2 / 4.25. Let me convert to fractions. 5.2 is 52/10, 4.25 is 17/4. So (52/10) divided by (17/4) is (52/10)*(4/17) = (208)/170 = 1.2235... Correct.

Debra: 2.8 / 5.6. That's 0.5. Correct.

Angela: 1.4 / 6.8. Multiply numerator and denominator by 10: 14 / 68 = 7 / 34 ≈ 0.2058. Correct.

So, yes, Evelyn's average speed is the highest.

But let me think again if there's another way to interpret the graph. For example, if the graph is showing the distance covered by each student over time, but each student's run is represented by a line, but the Asymptote code only draws dots. Wait, the Asymptote code draws dots at specific coordinates. So, maybe each student's run is represented by a single point, meaning that they all finished at different times, and the graph is showing their final position.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the problem is asking for the average speed during the entire run. However, without knowing the total time, we can't compute average speed.

But given that the problem is from a math competition or similar, and the answer is expected, the most plausible interpretation is that each student's dot represents their final position, and assuming they started at time 0, distance 0, so average speed is distance divided by time.

Therefore, Evelyn is the answer.

But let me check if there's another possible interpretation. For example, if the graph is showing the distance covered by each student at different times, but the question is about the instantaneous speed at a particular time? But the question says "average speed," not instantaneous.

Alternatively, maybe the graph is showing the distance covered by each student over time, and the slope of the line from the origin to their point is their average speed. Which is exactly what I calculated.

Yes, so if you plot a distance-time graph, the slope of the line from the origin to the point (time, distance) is the average speed. Therefore, the student with the steepest slope has the highest average speed.

So, calculating the slopes:

Evelyn: 4.5 / 1.25 = 3.6

Briana: 2.2 / 2.5 = 0.88

Carla: 5.2 / 4.25 ≈ 1.2235

Debra: 2.8 / 5.6 = 0.5

Angela: 1.4 / 6.8 ≈ 0.2059

So, Evelyn's slope is the steepest, so she has the highest average speed.

Therefore, the answer is Evelyn.

But let me check if the coordinates are (time, distance) or (distance, time). Wait, the axes are labeled: "distance" on the left (y-axis) and "time" on the bottom (x-axis). So, the coordinates are (time, distance). Therefore, the x-coordinate is time, y-coordinate is distance. Therefore, the slope is distance over time, which is speed. So, yes, the calculation is correct.

Therefore, Evelyn has the greatest average speed.

**Final Answer**
\boxed{Evelyn}

To determine which student has the greatest **average speed**, we analyze the given **distance-time graph**. Each student is represented by a single point on the graph, with coordinates $(x, y)$, where:

- $x$ represents **time** (in some unit),
- $y$ represents **distance** (in some unit).

---

### Understanding Average Speed

Average speed is defined as:

$$
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
$$

In a **distance-time graph**, the slope of the line from the origin $(0, 0)$ to a point $(t, d)$ gives the **average speed** of the student, since:

$$
\text{Slope} = \frac{d}{t}
$$

So, for each student, we calculate the slope of the line from the origin to their point on the graph.

---

### Calculating Average Speed for Each Student

Let’s compute the average speed for each student using their coordinates:

- **Evelyn**: $(1.25, 4.5)$
  $$
  \text{Speed} = \frac{4.5}{1.25} = 3.6
  $$

- **Briana**: $(2.5, 2.2)$
  $$
  \text{Speed} = \frac{2.2}{2.5} = 0.88
  $$

- **Carla**: $(4.25, 5.2)$
  $$
  \text{Speed} = \frac{5.2}{4.25} \approx 1.2235
  $$

- **Debra**: $(5.6, 2.8)$
  $$
  \text{Speed} = \frac{2.8}{5.6} = 0.5
  $$

- **Angela**: $(6.8, 1.4)$
  $$
  \text{Speed} = \frac{1.4}{6.8} \approx 0.2059
  $$

---

### Comparing the Speeds

Now, comparing the calculated average speeds:

- Evelyn: **3.6**
- Briana: **0.88**
- Carla: **1.2235**
- Debra: **0.5**
- Angela: **0.2059**

Clearly, **Evelyn** has the **highest average speed**.

---

### Final Answer

$$
\boxed{Evelyn}
$$

[Assistant]
Explanation:
Okay, so I need to figure out which student has the greatest average speed based on the graph provided. Let me start by recalling what average speed means. Average speed is calculated by dividing the total distance traveled by the total time taken. So, in this case, since the graph is a distance-time graph, the average speed for each student would be the slope of the line connecting their starting point to their ending point. Wait, but actually, since each student is represented by a single dot on the graph, does that mean they all started at the same time and ran until the same time? Or is each dot representing their position at a specific time?

Looking at the Asymptote code, there are dots at different coordinates. For example, Evelyn is at (1.25, 4.5), Briana at (2.5, 2.2), Carla at (4.25, 5.2), Debra at (5.6, 2.8), and Angela at (6.8, 1.4). The axes are labeled with time on the x-axis and distance on the y-axis. The graph has vertical lines from x=0 to x=8 and horizontal lines from y=0 to y=6. The labels for the axes are "distance" on the left (y-axis) and "time" on the bottom (x-axis).

So, each student's position is plotted at a specific time and distance. But wait, if they are just single points, how do we determine their average speed? Maybe the graph is showing the distance covered at different times, so each dot represents the distance covered by each student at a particular time. But if that's the case, then to find average speed, we need to know the total distance each student covered over the total time. However, the problem is that the graph doesn't show the starting point. Wait, but maybe all students started at time 0? If they all started at time 0, then their positions at time t would be their distance covered. But the problem is that the graph only shows one point for each student. So, unless they all finished at the same time, but the x-coordinate of each dot is different. For example, Evelyn is at x=1.25, Briana at x=2.5, Carla at x=4.25, Debra at x=5.6, and Angela at x=6.8. So, each student's dot is at a different time.

Wait, but if the graph is of a training run, maybe each student ran for a different amount of time, and the dot represents their position at the end of their run? If that's the case, then the average speed would be the distance divided by the time. So, for each student, their average speed is (distance)/time.

But then, how do we know the total distance? For example, Evelyn's dot is at (1.25, 4.5). So, that would mean at time 1.25 units, she had covered 4.5 units of distance. So, if we assume that she started at time 0, distance 0, then her average speed would be 4.5 / 1.25. Similarly for the others.

But wait, the problem says "the results of a cross-country team's training run are graphed below." So, maybe each student's run is represented by a single point? That seems odd. Usually, a distance-time graph would show the position over time, so if they are plotting a single point, maybe it's the final position? But then, if they all started at the same time, but different times? Wait, maybe the graph is showing the distance covered by each student at different times? But how can we know the total time each student ran?

Alternatively, maybe the graph is showing the distance covered by each student at the same time? But that would be a different interpretation.

Wait, let me think again. The Asymptote code draws a grid with vertical lines from x=0 to x=8 and horizontal lines from y=0 to y=6. The axes are labeled with time on the x-axis and distance on the y-axis. The dots are plotted at different coordinates. Each dot is labeled with the student's name. So, perhaps each dot represents the position of the student at a specific time. For example, Evelyn is at (1.25, 4.5), meaning that at time 1.25 units, she was at distance 4.5 units. Similarly, Briana is at (2.5, 2.2), so at time 2.5 units, she was at distance 2.2 units.

But if that's the case, then to find the average speed, we need to know the total distance each student covered over the total time they ran. However, the problem is that we don't know when each student started or when they stopped. If they all started at time 0, then their average speed would be their distance at the time they stopped divided by the time they stopped. But if they started at different times, we can't know.

Wait, but maybe the graph is showing the distance covered by each student at the end of their run. So, for example, Evelyn ran for 1.25 units of time and covered 4.5 units of distance. Then her average speed is 4.5 / 1.25. Similarly, Briana ran for 2.5 units of time and covered 2.2 units of distance, so her average speed is 2.2 / 2.5. Carla ran for 4.25 units of time and covered 5.2 units of distance, so 5.2 / 4.25. Debra ran for 5.6 units of time and covered 2.8 units of distance, so 2.8 / 5.6. Angela ran for 6.8 units of time and covered 1.4 units of distance, so 1.4 / 6.8.

If that's the case, then we can calculate each student's average speed and compare them.

Alternatively, maybe the graph is showing the distance covered at different times, but each student's run is represented by a single point. But that would not make sense for average speed unless we assume they started at time 0 and stopped at their respective times.

Alternatively, maybe the graph is showing the distance covered by each student at the same time? For example, at time t, each student's position is plotted. But then, if they all ran until the same time, but the graph only shows one point for each student, that would be their position at that time. But then, how do we know their total distance?

But given that the problem is asking for average speed, which is total distance over total time, and given that the graph is a distance-time graph, the most logical interpretation is that each student's dot represents their position at a specific time. If we assume that each student started at time 0, distance 0, then their average speed would be their distance divided by their time.

Alternatively, if they didn't start at time 0, but the graph is showing their position at some time, but without knowing their starting time, we can't calculate average speed. However, since the problem is presented in a way that expects an answer, it's likely that we are supposed to assume that each student started at time 0, and the dot represents their position at the end of their run. Therefore, their average speed is distance divided by time.

So, proceeding with that assumption, let's calculate each student's average speed.

First, let's note down each student's coordinates:

- Evelyn: (1.25, 4.5)
- Briana: (2.5, 2.2)
- Carla: (4.25, 5.2)
- Debra: (5.6, 2.8)
- Angela: (6.8, 1.4)

Assuming that the x-coordinate is time and the y-coordinate is distance, then:

For Evelyn: time = 1.25, distance = 4.5. Average speed = 4.5 / 1.25

Let me compute that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 1.25 times 3.6 is 4.5. So 4.5 / 1.25 = 3.6 units per time unit.

For Briana: time = 2.5, distance = 2.2. Average speed = 2.2 / 2.5. Let me calculate that. 2.2 divided by 2.5. Well, 2.5 goes into 2.2 0.88 times. So 0.88 units per time unit.

For Carla: time = 4.25, distance = 5.2. Average speed = 5.2 / 4.25. Let me compute that. 4.25 times 1.2 is 5.1, so 5.2 / 4.25 is approximately 1.2235 units per time unit.

For Debra: time = 5.6, distance = 2.8. Average speed = 2.8 / 5.6 = 0.5 units per time unit.

For Angela: time = 6.8, distance = 1.4. Average speed = 1.4 / 6.8. Let me calculate that. 1.4 divided by 6.8 is approximately 0.2059 units per time unit.

So, comparing all these:

Evelyn: 3.6

Briana: 0.88

Carla: ~1.22

Debra: 0.5

Angela: ~0.206

So, Evelyn has the highest average speed. Therefore, the answer should be Evelyn.

But wait, let me check if my interpretation is correct. Because if the graph is not showing the total distance covered by each student, but instead, maybe each student's position at a certain time, but they might have different starting times? For example, maybe some students started later? But the problem doesn't mention anything about that.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the question is about the average speed during the entire training run. However, without knowing the total time each student ran, we can't compute average speed.

But given that the problem is presented with a graph where each student is represented by a single point, and the axes are time and distance, the most logical assumption is that each student's run is represented by a single point, meaning that they all started at time 0, and the dot is their position at the end of their run. Therefore, their average speed is distance divided by time.

Alternatively, if the graph is showing the distance covered at different times, but each student's run is represented by a single point, maybe the graph is showing the distance covered at the same time? For example, at time t, each student's distance is plotted. But then, if they all ran until the same time, but the graph only shows one point for each student, then we would need to know their total distance. However, in that case, the average speed would be total distance divided by total time. But if the graph is showing their position at a specific time, then unless we know their starting point, we can't compute average speed.

But since the problem is from a cross-country team's training run, it's possible that each student ran for a different duration, and the graph is showing their final position. Therefore, assuming they all started at time 0, distance 0, then their average speed is distance divided by time.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the question is about the average speed during the entire run. However, without knowing the total time each student ran, we can't compute average speed.

But given that the problem is presented with the Asymptote code, which plots the points with specific coordinates, and given that the answer is expected, I think the first interpretation is correct.

Therefore, proceeding with that, Evelyn has the highest average speed of 3.6, so the answer is Evelyn.

But let me check again the calculations to make sure I didn't make any arithmetic errors.

Evelyn: 4.5 / 1.25. Let me do this division again. 1.25 times 3 is 3.75. 4.5 minus 3.75 is 0.75. 0.75 divided by 1.25 is 0.6. So total is 3.6. Correct.

Briana: 2.2 / 2.5. Multiply numerator and denominator by 2 to eliminate decimal: 4.4 / 5 = 0.88. Correct.

Carla: 5.2 / 4.25. Let me convert to fractions. 5.2 is 52/10, 4.25 is 17/4. So (52/10) divided by (17/4) is (52/10)*(4/17) = (208)/170 = 1.2235... Correct.

Debra: 2.8 / 5.6. That's 0.5. Correct.

Angela: 1.4 / 6.8. Multiply numerator and denominator by 10: 14 / 68 = 7 / 34 ≈ 0.2058. Correct.

So, yes, Evelyn's average speed is the highest.

But let me think again if there's another way to interpret the graph. For example, if the graph is showing the distance covered by each student over time, but each student's run is represented by a line, but the Asymptote code only draws dots. Wait, the Asymptote code draws dots at specific coordinates. So, maybe each student's run is represented by a single point, meaning that they all finished at different times, and the graph is showing their final position.

Alternatively, maybe the graph is showing the distance covered by each student at different times, but the problem is asking for the average speed during the entire run. However, without knowing the total time, we can't compute average speed.

But given that the problem is from a math competition or similar, and the answer is expected, the most plausible interpretation is that each student's dot represents their final position, and assuming they started at time 0, distance 0, so average speed is distance divided by time.

Therefore, Evelyn is the answer.

But let me check if there's another possible interpretation. For example, if the graph is showing the distance covered by each student at different times, but the question is about the instantaneous speed at a particular time? But the question says "average speed," not instantaneous.

Alternatively, maybe the graph is showing the distance covered by each student over time, and the slope of the line from the origin to their point is their average speed. Which is exactly what I calculated.

Yes, so if you plot a distance-time graph, the slope of the line from the origin to the point (time, distance) is the average speed. Therefore, the student with the steepest slope has the highest average speed.

So, calculating the slopes:

Evelyn: 4.5 / 1.25 = 3.6

Briana: 2.2 / 2.5 = 0.88

Carla: 5.2 / 4.25 ≈ 1.2235

Debra: 2.8 / 5.6 = 0.5

Angela: 1.4 / 6.8 ≈ 0.2059

So, Evelyn's slope is the steepest, so she has the highest average speed.

Therefore, the answer is Evelyn.

But let me check if the coordinates are (time, distance) or (distance, time). Wait, the axes are labeled: "distance" on the left (y-axis) and "time" on the bottom (x-axis). So, the coordinates are (time, distance). Therefore, the x-coordinate is time, y-coordinate is distance. Therefore, the slope is distance over time, which is speed. So, yes, the calculation is correct.

Therefore, Evelyn has the greatest average speed.

**Final Answer**
\boxed{Evelyn}

To determine which student has the greatest **average speed**, we analyze the given **distance-time graph**. Each student is represented by a single point on the graph, with coordinates $(x, y)$, where:

- $x$ represents **time** (in some unit),
- $y$ represents **distance** (in some unit).

---

### Understanding Average Speed

Average speed is defined as:

$$
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
$$

In a **distance-time graph**, the slope of the line from the origin $(0, 0)$ to a point $(t, d)$ gives the **average speed** of the student, since:

$$
\text{Slope} = \frac{d}{t}
$$

So, for each student, we calculate the slope of the line from the origin to their point on the graph.

---

### Calculating Average Speed for Each Student

Let’s compute the average speed for each student using their coordinates:

- **Evelyn**: $(1.25, 4.5)$
  $$
  \text{Speed} = \frac{4.5}{1.25} = 3.6
  $$

- **Briana**: $(2.5, 2.2)$
  $$
  \text{Speed} = \frac{2.2}{2.5} = 0.88
  $$

- **Carla**: $(4.25, 5.2)$
  $$
  \text{Speed} = \frac{5.2}{4.25} \approx 1.2235
  $$

- **Debra**: $(5.6, 2.8)$
  $$
  \text{Speed} = \frac{2.8}{5.6} = 0.5
  $$

- **Angela**: $(6.8, 1.4)$
  $$
  \text{Speed} = \frac{1.4}{6.8} \approx 0.2059
  $$

---

### Comparing the Speeds

Now, comparing the calculated average speeds:

- Evelyn: **3.6**
- Briana: **0.88**
- Carla: **1.2235**
- Debra: **0.5**
- Angela: **0.2059**

Clearly, **Evelyn** has the **highest average speed**.

---

### Final Answer

$$
\boxed{Evelyn}
$$

Solution: Evelyn covered more distance in less time than Briana, Debra and Angela, so her average speed is greater than any of their average speeds. Evelyn went almost as far as Carla in less than half the time that it took Carla, so Evelyn's average speed is also greater than Carla's. Therefore, $\boxed{\text{Evelyn}}$ is our answer.


Explanation:
Okay, so I need to find the perimeter of a regular hexagon that's divided into six equilateral triangles. They told me that the perimeter of one of those triangles is 21 inches. Let me think about how to approach this.

First, let me visualize a regular hexagon. A regular hexagon has six equal sides and six equal angles. If it's divided into six equilateral triangles, that probably means that each triangle has sides that are equal to the sides of the hexagon. Wait, how exactly is the hexagon divided into triangles?

I remember that a regular hexagon can be split into six equilateral triangles by drawing lines from the center to each of the six vertices. So each of those triangles would have two sides that are radii of the hexagon and one side that is a side of the hexagon. But if the triangles are equilateral, then all three sides must be equal. That means the radius of the hexagon is equal to the length of its side. So each triangle is equilateral with sides equal to the side length of the hexagon.

So if the perimeter of one of those triangles is 21 inches, then each side of the triangle is 21 divided by 3, right? Because a triangle has three sides. Let me write that down:

Perimeter of triangle = 3 * side length

So, side length of triangle = 21 / 3 = 7 inches.

Therefore, each side of the triangle is 7 inches. But since the triangles are formed by the sides of the hexagon and the radii, and since the triangles are equilateral, that means the side length of the hexagon is equal to the side length of the triangle. Therefore, each side of the hexagon is 7 inches.

Wait, but hold on. Let me confirm that. If the hexagon is divided into six equilateral triangles by connecting the center to each vertex, then each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. For the triangle to be equilateral, all three sides must be equal. Therefore, the radius of the hexagon must be equal to the length of its side. Therefore, the side length of the hexagon is equal to the radius. Therefore, each triangle has sides of length equal to the side of the hexagon. Therefore, if the perimeter of the triangle is 21, then each side is 7, so the side of the hexagon is 7. Therefore, the perimeter of the hexagon is 6 times 7, which is 42 inches. Is that right?

Wait, but let me check again. Maybe I made a mistake here. Let me think again. Let me recall some properties of regular hexagons. A regular hexagon can be divided into six equilateral triangles, each with side length equal to the side length of the hexagon. Therefore, if each triangle has a perimeter of 21, then each side of the triangle is 7, so the side of the hexagon is 7. Therefore, the perimeter of the hexagon is 6*7=42. So the answer is 42 inches?

But let me think again. Maybe there's a different way to divide the hexagon into triangles? Wait, the problem says "a regular hexagon can be divided into six equilateral triangles." So that division is possible. So if you divide it into six equilateral triangles, then each triangle has sides equal to the side of the hexagon. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if the perimeter of the triangle is 21, then the side length of the hexagon is 7, so the perimeter of the hexagon is 6*7=42. So that seems straightforward.

But maybe I need to check if the triangles are not the ones formed by connecting the center to the vertices. Maybe there's another way to divide the hexagon into six equilateral triangles? For example, maybe dividing it into smaller triangles? But the problem says "a regular hexagon can be divided into six equilateral triangles," so probably the standard way, which is connecting the center to each vertex. So each triangle is equilateral. Therefore, the side length of the triangle is equal to the side length of the hexagon. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if that perimeter is 21, then the side length is 7, so the perimeter of the hexagon is 6*7=42. Therefore, the answer is 42 inches.

But let me think again. Maybe the triangles are not the ones with the center? Maybe the problem is referring to a different division? For example, if you divide the hexagon into six equilateral triangles by drawing lines from one vertex to all the others? Wait, but that would create different triangles. Let me think. If you take a regular hexagon and draw lines from one vertex to all the non-adjacent vertices, but I think that would create different shapes. Maybe not equilateral triangles. Let me try to imagine.

Alternatively, maybe the problem is referring to dividing the hexagon into six equilateral triangles in some other way. But I think the standard division is connecting the center to each vertex, creating six equilateral triangles. So each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. For those triangles to be equilateral, the radius must be equal to the side length of the hexagon. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, perimeter of triangle is 3 times side length. Therefore, if perimeter is 21, then side length is 7. Therefore, perimeter of hexagon is 6*7=42. Therefore, answer is 42.

But maybe I need to check with actual formulas. Let me recall that in a regular hexagon, the side length is equal to the radius of the circumscribed circle. So if you connect the center to each vertex, the distance from the center to any vertex is equal to the side length. Therefore, each of those triangles is equilateral. Therefore, the perimeter of each triangle is 3*s, where s is the side length of the hexagon. Therefore, given that perimeter is 21, then s=7. Therefore, perimeter of hexagon is 6*s=42. Therefore, answer is 42. So I think that is correct.

But let me check again. Suppose the side length of the hexagon is s. Then each triangle has sides of length s, s, and s. Therefore, perimeter is 3s. Therefore, 3s=21, so s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42. So that seems straightforward.

Alternatively, maybe the problem is trying to trick me? Let me think. Suppose that the triangles are not the ones with the center. Suppose that the hexagon is divided into six equilateral triangles in some other way. For example, maybe dividing it into smaller triangles? But if you divide a regular hexagon into six equilateral triangles, the most straightforward way is connecting the center. If you divide it in another way, maybe the triangles would have different side lengths. But the problem says "a regular hexagon can be divided into six equilateral triangles." So that division is possible, and the problem is referring to that division. Therefore, the answer is 42.

Alternatively, maybe the problem is referring to the triangles formed by splitting the hexagon into six congruent equilateral triangles. But again, that would be the same as connecting the center. Therefore, I think my answer is correct.

Wait, but let me think again. Let me check with actual measurements. Let me take a regular hexagon with side length 7. Then, connecting the center to each vertex divides it into six equilateral triangles. Each triangle has sides of length 7. Therefore, perimeter of each triangle is 21. Therefore, that matches the problem's statement. Therefore, the perimeter of the hexagon is 6*7=42. Therefore, the answer is 42. Therefore, I think that is correct.

But maybe I need to check if the triangles are not equilateral? Wait, the problem says that the hexagon can be divided into six equilateral triangles. Therefore, they are equilateral. Therefore, each triangle has three sides equal. Therefore, each side of the triangle is equal to the side of the hexagon. Therefore, the perimeter of the triangle is 3 times the side of the hexagon. Therefore, given that perimeter is 21, then side of hexagon is 7. Therefore, perimeter of hexagon is 6*7=42. Therefore, answer is 42. So I think that is correct.

Alternatively, maybe the problem is referring to a different kind of division? For example, if you divide the hexagon into six equilateral triangles by drawing lines from the midpoints of the sides? But that would not result in equilateral triangles. Let me think. If you take a regular hexagon and connect midpoints, maybe you get different shapes. For example, connecting midpoints might create smaller hexagons or other polygons, but not necessarily equilateral triangles. Therefore, I think the standard division is connecting the center to the vertices. Therefore, the answer is 42.

Alternatively, maybe the problem is in three dimensions? No, the problem is about a regular hexagon, which is two-dimensional. Therefore, I think my answer is correct.

Therefore, I think the perimeter of the regular hexagon is 42 inches.

**Final Answer**
The perimeter of the regular hexagon is \boxed{42} inches.

To solve the problem, we begin by understanding the geometric relationship between a regular hexagon and the equilateral triangles it can be divided into.

---

### Step 1: Understanding the Division of the Hexagon

A **regular hexagon** has six equal sides and six equal angles. One of the standard ways to divide it is by drawing lines from the **center** of the hexagon to each of its **six vertices**. This creates **six equilateral triangles**, each with:

- Two sides that are **radii** of the hexagon (from the center to a vertex),
- One side that is a **side of the hexagon**.

Since the hexagon is regular, all of its sides and angles are equal, and the radii from the center to the vertices are also equal in length. Therefore, each of these six triangles is an **equilateral triangle**, with all three sides equal in length.

---

### Step 2: Relating Triangle Perimeter to Hexagon Side Length

We are told that the **perimeter of one of these equilateral triangles is 21 inches**. Since an equilateral triangle has three equal sides, we can find the length of one side as:

$$
\text{Side length of triangle} = \frac{\text{Perimeter of triangle}}{3} = \frac{21}{3} = 7 \text{ inches}
$$

Because the triangle is equilateral and formed by the side of the hexagon and two radii, the **side length of the triangle is equal to the side length of the hexagon**. Therefore, the **side length of the hexagon is also 7 inches**.

---

### Step 3: Calculating the Perimeter of the Hexagon

A regular hexagon has **six equal sides**, so its perimeter is:

$$
\text{Perimeter of hexagon} = 6 \times \text{side length} = 6 \times 7 = 42 \text{ inches}
$$

---

### Final Answer

$$
\boxed{42}
$$

[Assistant]
Explanation:
Okay, so I need to find the perimeter of a regular hexagon that's divided into six equilateral triangles. They told me that the perimeter of one of those triangles is 21 inches. Let me think about how to approach this.

First, let me visualize a regular hexagon. A regular hexagon has six equal sides and six equal angles. If it's divided into six equilateral triangles, that probably means that each triangle has sides that are equal to the sides of the hexagon. Wait, how exactly is the hexagon divided into triangles?

I remember that a regular hexagon can be split into six equilateral triangles by drawing lines from the center to each of the six vertices. So each of those triangles would have two sides that are radii of the hexagon and one side that is a side of the hexagon. But if the triangles are equilateral, then all three sides must be equal. That means the radius of the hexagon is equal to the length of its side. So each triangle is equilateral with sides equal to the side length of the hexagon.

So if the perimeter of one of those triangles is 21 inches, then each side of the triangle is 21 divided by 3, right? Because a triangle has three sides. Let me write that down:

Perimeter of triangle = 3 * side length

So, side length of triangle = 21 / 3 = 7 inches.

Therefore, each side of the triangle is 7 inches. But since the triangles are formed by the sides of the hexagon and the radii, and since the triangles are equilateral, that means the side length of the hexagon is equal to the side length of the triangle. Therefore, each side of the hexagon is 7 inches.

Wait, but hold on. Let me confirm that. If the hexagon is divided into six equilateral triangles by connecting the center to each vertex, then each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. For the triangle to be equilateral, all three sides must be equal. Therefore, the radius of the hexagon must be equal to the length of its side. Therefore, the side length of the hexagon is equal to the radius. Therefore, each triangle has sides of length equal to the side of the hexagon. Therefore, if the perimeter of the triangle is 21, then each side is 7, so the side of the hexagon is 7. Therefore, the perimeter of the hexagon is 6 times 7, which is 42 inches. Is that right?

Wait, but let me check again. Maybe I made a mistake here. Let me think again. Let me recall some properties of regular hexagons. A regular hexagon can be divided into six equilateral triangles, each with side length equal to the side length of the hexagon. Therefore, if each triangle has a perimeter of 21, then each side of the triangle is 7, so the side of the hexagon is 7. Therefore, the perimeter of the hexagon is 6*7=42. So the answer is 42 inches?

But let me think again. Maybe there's a different way to divide the hexagon into triangles? Wait, the problem says "a regular hexagon can be divided into six equilateral triangles." So that division is possible. So if you divide it into six equilateral triangles, then each triangle has sides equal to the side of the hexagon. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if the perimeter of the triangle is 21, then the side length of the hexagon is 7, so the perimeter of the hexagon is 6*7=42. So that seems straightforward.

But maybe I need to check if the triangles are not the ones formed by connecting the center to the vertices. Maybe there's another way to divide the hexagon into six equilateral triangles? For example, maybe dividing it into smaller triangles? But the problem says "a regular hexagon can be divided into six equilateral triangles," so probably the standard way, which is connecting the center to each vertex. So each triangle is equilateral. Therefore, the side length of the triangle is equal to the side length of the hexagon. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if that perimeter is 21, then the side length is 7, so the perimeter of the hexagon is 6*7=42. Therefore, the answer is 42 inches.

But let me think again. Maybe the triangles are not the ones with the center? Maybe the problem is referring to a different division? For example, if you divide the hexagon into six equilateral triangles by drawing lines from one vertex to all the others? Wait, but that would create different triangles. Let me think. If you take a regular hexagon and draw lines from one vertex to all the non-adjacent vertices, but I think that would create different shapes. Maybe not equilateral triangles. Let me try to imagine.

Alternatively, maybe the problem is referring to dividing the hexagon into six equilateral triangles in some other way. But I think the standard division is connecting the center to each vertex, creating six equilateral triangles. So each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. For those triangles to be equilateral, the radius must be equal to the side length of the hexagon. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, perimeter of triangle is 3 times side length. Therefore, if perimeter is 21, then side length is 7. Therefore, perimeter of hexagon is 6*7=42. Therefore, answer is 42.

But maybe I need to check with actual formulas. Let me recall that in a regular hexagon, the side length is equal to the radius of the circumscribed circle. So if you connect the center to each vertex, the distance from the center to any vertex is equal to the side length. Therefore, each of those triangles is equilateral. Therefore, the perimeter of each triangle is 3*s, where s is the side length of the hexagon. Therefore, given that perimeter is 21, then s=7. Therefore, perimeter of hexagon is 6*s=42. Therefore, answer is 42. So I think that is correct.

But let me check again. Suppose the side length of the hexagon is s. Then each triangle has sides of length s, s, and s. Therefore, perimeter is 3s. Therefore, 3s=21, so s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42. So that seems straightforward.

Alternatively, maybe the problem is trying to trick me? Let me think. Suppose that the triangles are not the ones with the center. Suppose that the hexagon is divided into six equilateral triangles in some other way. For example, maybe dividing it into smaller triangles? But if you divide a regular hexagon into six equilateral triangles, the most straightforward way is connecting the center. If you divide it in another way, maybe the triangles would have different side lengths. But the problem says "a regular hexagon can be divided into six equilateral triangles." So that division is possible, and the problem is referring to that division. Therefore, the answer is 42.

Alternatively, maybe the problem is referring to the triangles formed by splitting the hexagon into six congruent equilateral triangles. But again, that would be the same as connecting the center. Therefore, I think my answer is correct.

Wait, but let me think again. Let me check with actual measurements. Let me take a regular hexagon with side length 7. Then, connecting the center to each vertex divides it into six equilateral triangles. Each triangle has sides of length 7. Therefore, perimeter of each triangle is 21. Therefore, that matches the problem's statement. Therefore, the perimeter of the hexagon is 6*7=42. Therefore, the answer is 42. Therefore, I think that is correct.

But maybe I need to check if the triangles are not equilateral? Wait, the problem says that the hexagon can be divided into six equilateral triangles. Therefore, they are equilateral. Therefore, each triangle has three sides equal. Therefore, each side of the triangle is equal to the side of the hexagon. Therefore, the perimeter of the triangle is 3 times the side of the hexagon. Therefore, given that perimeter is 21, then side of hexagon is 7. Therefore, perimeter of hexagon is 6*7=42. Therefore, answer is 42. So I think that is correct.

Alternatively, maybe the problem is referring to a different kind of division? For example, if you divide the hexagon into six equilateral triangles by drawing lines from the midpoints of the sides? But that would not result in equilateral triangles. Let me think. If you take a regular hexagon and connect midpoints, maybe you get different shapes. For example, connecting midpoints might create smaller hexagons or other polygons, but not necessarily equilateral triangles. Therefore, I think the standard division is connecting the center to the vertices. Therefore, the answer is 42.

Alternatively, maybe the problem is in three dimensions? No, the problem is about a regular hexagon, which is two-dimensional. Therefore, I think my answer is correct.

Therefore, I think the perimeter of the regular hexagon is 42 inches.

**Final Answer**
The perimeter of the regular hexagon is \boxed{42} inches.

To solve the problem, we begin by understanding the geometric relationship between a regular hexagon and the equilateral triangles it can be divided into.

---

### Step 1: Understanding the Division of the Hexagon

A **regular hexagon** has six equal sides and six equal angles. One of the standard ways to divide it is by drawing lines from the **center** of the hexagon to each of its **six vertices**. This creates **six equilateral triangles**, each with:

- Two sides that are **radii** of the hexagon (from the center to a vertex),
- One side that is a **side of the hexagon**.

Since the hexagon is regular, all of its sides and angles are equal, and the radii from the center to the vertices are also equal in length. Therefore, each of these six triangles is an **equilateral triangle**, with all three sides equal in length.

---

### Step 2: Relating Triangle Perimeter to Hexagon Side Length

We are told that the **perimeter of one of these equilateral triangles is 21 inches**. Since an equilateral triangle has three equal sides, we can find the length of one side as:

$$
\text{Side length of triangle} = \frac{\text{Perimeter of triangle}}{3} = \frac{21}{3} = 7 \text{ inches}
$$

Because the triangle is equilateral and formed by the side of the hexagon and two radii, the **side length of the triangle is equal to the side length of the hexagon**. Therefore, the **side length of the hexagon is also 7 inches**.

---

### Step 3: Calculating the Perimeter of the Hexagon

A regular hexagon has **six equal sides**, so its perimeter is:

$$
\text{Perimeter of hexagon} = 6 \times \text{side length} = 6 \times 7 = 42 \text{ inches}
$$

---

### Final Answer

$$
\boxed{42}
$$

Solution:
The side length of the hexagon is equal to the side length of one of the equilateral triangles.  Since the hexagon has six sides and the triangle has three sides, the perimeter of the hexagon is twice as large as the perimeter of a triangle.  Therefore, the perimeter of the hexagon is $2(21\text{ inches})=\boxed{42}$ inches.

[asy]
unitsize(10mm);
defaultpen(linewidth(.7pt)+fontsize(8pt));

int i;

for(i=1;i<=6;++i)
{

draw(dir(60*i)--dir(60*(i+1)));
}

for(i=1;i<=3;++i)
{

draw(dir(60*i)--dir(60*(i+3)));
}[/asy]


Explanation:
Okay, so I need to find the height of a cylinder when the volume is given as 45π cubic centimeters. Let me recall the formula for the volume of a cylinder. I think it's V equals π times radius squared times height. Let me write that down to make sure:

V = πr²h

Alright, so they told me the volume is 45π, so I can set up the equation:

45π = πr²h

Hmm, okay. Now, I need to find the height, which is h. But wait, do I know the radius? Let me check the Asymptote code they provided. The Asymptote code draws a cylinder with some labels. Let me try to visualize it. There's a label that says "r=3" near the top, between the points (2.2,0) and (3.4,0). So that must mean the radius is 3 centimeters. Let me confirm that.

In the Asymptote code, they draw a circle that's shifted and scaled. The original circle is at (0,0) with radius 1.2, but then it's scaled by yscale(0.3), which might be a vertical scaling. However, the label says "r=3", so maybe the radius is 3 cm. Let me think. The Asymptote code draws a circle with radius 1.2, but then it's scaled vertically by 0.3. However, the label for the radius is placed between (2.2,0) and (3.4,0), which is a horizontal distance. The length between 2.2 and 3.4 is 3.4 - 2.2 = 1.2 units. But the label says "r=3", so maybe the actual radius is 3 cm, and the Asymptote code is just using a scaled drawing. So maybe the radius is 3 cm. Let me go with that because the problem mentions the radius is 3 cm. So r = 3 cm.

So plugging that into the volume formula:

45π = π*(3)²*h

Let me compute 3 squared. 3 squared is 9. So:

45π = π*9*h

Now, I can divide both sides of the equation by π to simplify:

45 = 9h

Then, to solve for h, divide both sides by 9:

h = 45 / 9

Which is 5. So h equals 5 centimeters. Let me check my steps again to make sure I didn't make a mistake.

First, volume formula: correct. Then, substituting the given volume and radius: yes, 45π equals πr²h. Then, radius is 3, so squared is 9. Dividing both sides by π: 45 = 9h. Dividing by 9: h = 5. Seems straightforward.

Wait, but let me check if the radius is indeed 3. The Asymptote code might have some scaling. Let me think again. The Asymptote code draws a circle with radius 1.2, but then applies a yscale(0.3). So vertically, the circle is compressed by a factor of 0.3, but horizontally, it's not scaled. Then, the label for the radius is between (2.2,0) and (3.4,0). The distance between those two points is 3.4 - 2.2 = 1.2 units. But in the drawing, the circle is shifted to (2.2, 0), so maybe the horizontal length of 1.2 units corresponds to the diameter? Wait, if the radius is 3 cm, then the diameter would be 6 cm. But in the drawing, the horizontal distance between the two points is 1.2 units. However, since the Asymptote code is just a diagram, the actual measurements might not correspond to the units in the code. The problem says "the cylinder shown" and gives the label "r=3", so I think we can safely assume that the radius is 3 cm. The Asymptote code is just for illustration, and the label explicitly states the radius is 3. Therefore, my calculation is correct.

Therefore, the height is 5 centimeters. Let me just verify once more:

Volume = πr²h = π*(3)^2*5 = π*9*5 = 45π. Yep, that matches the given volume. So the height is indeed 5 cm.

I think that's all. No steps seem to be missing, and the answer checks out. So I'm confident that the height is 5 centimeters.

**Final Answer**
The height of the cylinder is \boxed{5} centimeters.

To determine the height of the cylinder, we begin with the standard formula for the volume of a cylinder:

$$
V = \pi r^2 h
$$

We are given:
- The volume $ V = 45\pi $ cubic centimeters
- The radius $ r = 3 $ centimeters (as explicitly labeled in the diagram)

---

### Step 1: Plug in the known values

$$
45\pi = \pi (3)^2 h
$$

$$
45\pi = \pi \cdot 9 \cdot h
$$

---

### Step 2: Simplify the equation

Divide both sides of the equation by $ \pi $ to eliminate it:

$$
45 = 9h
$$

---

### Step 3: Solve for $ h $

$$
h = \frac{45}{9} = 5
$$

---

### Final Answer

The height of the cylinder is:

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to find the height of a cylinder when the volume is given as 45π cubic centimeters. Let me recall the formula for the volume of a cylinder. I think it's V equals π times radius squared times height. Let me write that down to make sure:

V = πr²h

Alright, so they told me the volume is 45π, so I can set up the equation:

45π = πr²h

Hmm, okay. Now, I need to find the height, which is h. But wait, do I know the radius? Let me check the Asymptote code they provided. The Asymptote code draws a cylinder with some labels. Let me try to visualize it. There's a label that says "r=3" near the top, between the points (2.2,0) and (3.4,0). So that must mean the radius is 3 centimeters. Let me confirm that.

In the Asymptote code, they draw a circle that's shifted and scaled. The original circle is at (0,0) with radius 1.2, but then it's scaled by yscale(0.3), which might be a vertical scaling. However, the label says "r=3", so maybe the radius is 3 cm. Let me think. The Asymptote code draws a circle with radius 1.2, but then it's scaled vertically by 0.3. However, the label for the radius is placed between (2.2,0) and (3.4,0), which is a horizontal distance. The length between 2.2 and 3.4 is 3.4 - 2.2 = 1.2 units. But the label says "r=3", so maybe the actual radius is 3 cm, and the Asymptote code is just using a scaled drawing. So maybe the radius is 3 cm. Let me go with that because the problem mentions the radius is 3 cm. So r = 3 cm.

So plugging that into the volume formula:

45π = π*(3)²*h

Let me compute 3 squared. 3 squared is 9. So:

45π = π*9*h

Now, I can divide both sides of the equation by π to simplify:

45 = 9h

Then, to solve for h, divide both sides by 9:

h = 45 / 9

Which is 5. So h equals 5 centimeters. Let me check my steps again to make sure I didn't make a mistake.

First, volume formula: correct. Then, substituting the given volume and radius: yes, 45π equals πr²h. Then, radius is 3, so squared is 9. Dividing both sides by π: 45 = 9h. Dividing by 9: h = 5. Seems straightforward.

Wait, but let me check if the radius is indeed 3. The Asymptote code might have some scaling. Let me think again. The Asymptote code draws a circle with radius 1.2, but then applies a yscale(0.3). So vertically, the circle is compressed by a factor of 0.3, but horizontally, it's not scaled. Then, the label for the radius is between (2.2,0) and (3.4,0). The distance between those two points is 3.4 - 2.2 = 1.2 units. But in the drawing, the circle is shifted to (2.2, 0), so maybe the horizontal length of 1.2 units corresponds to the diameter? Wait, if the radius is 3 cm, then the diameter would be 6 cm. But in the drawing, the horizontal distance between the two points is 1.2 units. However, since the Asymptote code is just a diagram, the actual measurements might not correspond to the units in the code. The problem says "the cylinder shown" and gives the label "r=3", so I think we can safely assume that the radius is 3 cm. The Asymptote code is just for illustration, and the label explicitly states the radius is 3. Therefore, my calculation is correct.

Therefore, the height is 5 centimeters. Let me just verify once more:

Volume = πr²h = π*(3)^2*5 = π*9*5 = 45π. Yep, that matches the given volume. So the height is indeed 5 cm.

I think that's all. No steps seem to be missing, and the answer checks out. So I'm confident that the height is 5 centimeters.

**Final Answer**
The height of the cylinder is \boxed{5} centimeters.

To determine the height of the cylinder, we begin with the standard formula for the volume of a cylinder:

$$
V = \pi r^2 h
$$

We are given:
- The volume $ V = 45\pi $ cubic centimeters
- The radius $ r = 3 $ centimeters (as explicitly labeled in the diagram)

---

### Step 1: Plug in the known values

$$
45\pi = \pi (3)^2 h
$$

$$
45\pi = \pi \cdot 9 \cdot h
$$

---

### Step 2: Simplify the equation

Divide both sides of the equation by $ \pi $ to eliminate it:

$$
45 = 9h
$$

---

### Step 3: Solve for $ h $

$$
h = \frac{45}{9} = 5
$$

---

### Final Answer

The height of the cylinder is:

$$
\boxed{5}
$$

Solution: The volume of the cylinder is $bh=\pi r^2h$. The radius of the base is $3$ cm, so we have $9\pi h=45\pi\qquad\Rightarrow h=5$. The height of the cylinder is $\boxed{5}$ cm.


Explanation:
Okay, let's see. The math club is having a bake sale to raise money for a trip. They sold some cookies, cupcakes, and brownies, and they have some costs. I need to figure out their profit. Profit is total revenue minus total costs, right? So first, I need to calculate how much money they made from selling all these items, and then subtract the cost of baking them. Let me break it down step by step.

First, let's handle the cookies. They sold 54 cookies at three for $1. Hmm, so three cookies cost $1. So, how many sets of three cookies did they sell? Let me divide the total number of cookies by 3. 54 divided by 3 is 18. So, they sold 18 sets of three cookies. Each set is $1, so the total revenue from cookies is 18 times $1, which is $18. Let me write that down: cookies revenue = 54 / 3 * 1 = $18.

Next, the cupcakes. They sold 20 cupcakes at $2 each. That seems straightforward. So, 20 times $2. Let me calculate that. 20 * 2 is $40. So cupcakes revenue is $40.

Then, the brownies. They sold 35 brownies at $1 each. So that's just 35 * 1, which is $35. So brownies revenue is $35.

Now, let me add up all the revenues from each item to get the total revenue. So that's $18 (cookies) + $40 (cupcakes) + $35 (brownies). Let me add them step by step. 18 + 40 is 58, and then 58 + 35 is 93. So total revenue is $93.

Now, the cost to bake these items was $15. So profit is total revenue minus total cost. That would be $93 - $15. Let me subtract that. 93 minus 15 is 78. So the profit is $78?

Wait, let me check my calculations again to make sure I didn't make a mistake. Sometimes when you do mental math, you might slip up.

Starting with cookies: 54 cookies at three for a dollar. 54 divided by 3 is indeed 18. 18 times $1 is $18. That seems right.

Cupcakes: 20 at $2 each. 20*2 is 40. Correct.

Brownies: 35 at $1 each. 35*1 is 35. Correct.

Total revenue: 18 + 40 + 35. Let me add 18 + 40 first. 18 + 40 is 58. Then 58 + 35. 58 + 30 is 88, plus 5 is 93. Yes, that's right.

Total cost is $15. So 93 - 15. Let me do that again. 93 - 10 is 83, minus 5 more is 78. Yep, $78. So the profit is $78.

Wait, but let me think again. Is there any chance that the cost is per item or something else? The problem says "it cost the math club $15 to bake these items." So that's the total cost, not per item. So subtracting $15 from total revenue is correct. So yes, $93 total revenue, minus $15 cost, profit is $78.

Hmm, seems straightforward. Let me check if I interpreted the prices correctly. For the cookies, three for $1. So each cookie is $1/3, but since they are sold in sets of three, the calculation is correct as 54 divided by 3 gives the number of sets, each set is $1. So that's right.

Alternatively, if I wanted to calculate per cookie, it would be 54 cookies * (1/3) dollars per cookie, which is 54/3 = 18 dollars. Same result. So that's correct.

Cupcakes and brownies are straightforward.

So, I think my answer is correct. The profit is $78.

**Final Answer**
The math club's profit was \boxed{78} dollars.

To determine the profit of the math club's bake sale, we need to follow a clear process of calculating total revenue and then subtracting the total cost.

---

### **Step 1: Calculate Revenue from Each Item**

#### **Cookies**
- Sold: 54 cookies
- Price: 3 cookies for $1
- Number of sets of 3 cookies:
  $$
  \frac{54}{3} = 18
  $$
- Revenue from cookies:
  $$
  18 \times 1 = 18 \text{ dollars}
  $$

#### **Cupcakes**
- Sold: 20 cupcakes
- Price: $2 each
- Revenue from cupcakes:
  $$
  20 \times 2 = 40 \text{ dollars}
  $$

#### **Brownies**
- Sold: 35 brownies
- Price: $1 each
- Revenue from brownies:
  $$
  35 \times 1 = 35 \text{ dollars}
  $$

---

### **Step 2: Calculate Total Revenue**

Add up the revenue from all items:
$$
18 + 40 + 35 = 93 \text{ dollars}
$$

---

### **Step 3: Subtract Total Cost**

The total cost to bake all the items was:
$$
15 \text{ dollars}
$$

---

### **Step 4: Calculate Profit**

Profit is calculated as:
$$
\text{Profit} = \text{Total Revenue} - \text{Total Cost}
$$
$$
\text{Profit} = 93 - 15 = 78 \text{ dollars}
$$

---

### **Final Answer**

$$
\boxed{78}
$$

[Assistant]
Explanation:
Okay, let's see. The math club is having a bake sale to raise money for a trip. They sold some cookies, cupcakes, and brownies, and they have some costs. I need to figure out their profit. Profit is total revenue minus total costs, right? So first, I need to calculate how much money they made from selling all these items, and then subtract the cost of baking them. Let me break it down step by step.

First, let's handle the cookies. They sold 54 cookies at three for $1. Hmm, so three cookies cost $1. So, how many sets of three cookies did they sell? Let me divide the total number of cookies by 3. 54 divided by 3 is 18. So, they sold 18 sets of three cookies. Each set is $1, so the total revenue from cookies is 18 times $1, which is $18. Let me write that down: cookies revenue = 54 / 3 * 1 = $18.

Next, the cupcakes. They sold 20 cupcakes at $2 each. That seems straightforward. So, 20 times $2. Let me calculate that. 20 * 2 is $40. So cupcakes revenue is $40.

Then, the brownies. They sold 35 brownies at $1 each. So that's just 35 * 1, which is $35. So brownies revenue is $35.

Now, let me add up all the revenues from each item to get the total revenue. So that's $18 (cookies) + $40 (cupcakes) + $35 (brownies). Let me add them step by step. 18 + 40 is 58, and then 58 + 35 is 93. So total revenue is $93.

Now, the cost to bake these items was $15. So profit is total revenue minus total cost. That would be $93 - $15. Let me subtract that. 93 minus 15 is 78. So the profit is $78?

Wait, let me check my calculations again to make sure I didn't make a mistake. Sometimes when you do mental math, you might slip up.

Starting with cookies: 54 cookies at three for a dollar. 54 divided by 3 is indeed 18. 18 times $1 is $18. That seems right.

Cupcakes: 20 at $2 each. 20*2 is 40. Correct.

Brownies: 35 at $1 each. 35*1 is 35. Correct.

Total revenue: 18 + 40 + 35. Let me add 18 + 40 first. 18 + 40 is 58. Then 58 + 35. 58 + 30 is 88, plus 5 is 93. Yes, that's right.

Total cost is $15. So 93 - 15. Let me do that again. 93 - 10 is 83, minus 5 more is 78. Yep, $78. So the profit is $78.

Wait, but let me think again. Is there any chance that the cost is per item or something else? The problem says "it cost the math club $15 to bake these items." So that's the total cost, not per item. So subtracting $15 from total revenue is correct. So yes, $93 total revenue, minus $15 cost, profit is $78.

Hmm, seems straightforward. Let me check if I interpreted the prices correctly. For the cookies, three for $1. So each cookie is $1/3, but since they are sold in sets of three, the calculation is correct as 54 divided by 3 gives the number of sets, each set is $1. So that's right.

Alternatively, if I wanted to calculate per cookie, it would be 54 cookies * (1/3) dollars per cookie, which is 54/3 = 18 dollars. Same result. So that's correct.

Cupcakes and brownies are straightforward.

So, I think my answer is correct. The profit is $78.

**Final Answer**
The math club's profit was \boxed{78} dollars.

To determine the profit of the math club's bake sale, we need to follow a clear process of calculating total revenue and then subtracting the total cost.

---

### **Step 1: Calculate Revenue from Each Item**

#### **Cookies**
- Sold: 54 cookies
- Price: 3 cookies for $1
- Number of sets of 3 cookies:
  $$
  \frac{54}{3} = 18
  $$
- Revenue from cookies:
  $$
  18 \times 1 = 18 \text{ dollars}
  $$

#### **Cupcakes**
- Sold: 20 cupcakes
- Price: $2 each
- Revenue from cupcakes:
  $$
  20 \times 2 = 40 \text{ dollars}
  $$

#### **Brownies**
- Sold: 35 brownies
- Price: $1 each
- Revenue from brownies:
  $$
  35 \times 1 = 35 \text{ dollars}
  $$

---

### **Step 2: Calculate Total Revenue**

Add up the revenue from all items:
$$
18 + 40 + 35 = 93 \text{ dollars}
$$

---

### **Step 3: Subtract Total Cost**

The total cost to bake all the items was:
$$
15 \text{ dollars}
$$

---

### **Step 4: Calculate Profit**

Profit is calculated as:
$$
\text{Profit} = \text{Total Revenue} - \text{Total Cost}
$$
$$
\text{Profit} = 93 - 15 = 78 \text{ dollars}
$$

---

### **Final Answer**

$$
\boxed{78}
$$

Solution:
To find the profit, we want to find out how much the math club earned from selling the various baked goods and subtract the cost of producing those goods, $\$15$, from the number we get.

First let's calculate how much the math club earned from selling cookies. The cookies were sold at a price of three for $\$1$, so the math club earned $54\div 3\cdot\$1=18\cdot\$1=\$18$ from selling cookies.

Next, let's calculate how much the club earned from selling cupcakes. At a price of $\$2$ each, the club earned $20\cdot \$2=\$40$ from selling cupcakes.

Finally, let's calculate how much the club earned from selling brownies. At a price of $\$1$ each, the club earned $35\cdot\$1=\$35$ from selling brownies.

Now let's add up these numbers to find out how much the club earned in total and subtract $\$15$ from that number to find the club's profit. We obtain \begin{align*}
\$18+\$40+\$35-\$15&=\$18+\$40+\$35-\$15\\
&=\$18+\$40+\$35+(-\$15)\\
&=\$18+\$40+(\$35+(-\$15))\\
&=\$18+\$40+(\$20)\\
&=\boxed{78}.
\end{align*}Notice how we used the definition of subtraction, $a-b=a+(-b)$ to $\$35-\$15$ as $\$35+(-\$15)$ and the associative property of addition to group the numbers together.


Explanation:
Okay, so I need to figure out what happens when I rotate the complex number 7 + 2i by 90 degrees counter-clockwise around the origin. Hmm, complex numbers and rotations... I remember that rotations in the complex plane can be done using multiplication by another complex number. Let me recall...

I think a rotation by an angle θ is achieved by multiplying the complex number by e^(iθ). Since we're dealing with a 90-degree rotation, θ is 90 degrees. But wait, in radians, that's π/2. So, e^(iπ/2) is... what's that? Euler's formula says e^(iθ) = cosθ + i sinθ. So cos(π/2) is 0 and sin(π/2) is 1. Therefore, e^(iπ/2) is 0 + i*1 = i.

So, multiplying the original complex number by i should give me the rotated one? Let me check. If I have a complex number z = a + bi, multiplying by i gives z*i = (a + bi)*i = ai + bi^2 = ai - b, since i^2 is -1. So that's -b + ai. Which is like rotating the point (a, b) 90 degrees counter-clockwise. Let me visualize that. If I have a point (a, b), rotating it 90 degrees CCW would take it to (-b, a). Yes, that matches with the result of multiplying by i. So, for example, if I have 1 + 0i, multiplying by i gives 0 + 1i, which is (0,1), which is indeed a 90-degree rotation from (1,0). So that seems right.

Therefore, applying this to 7 + 2i. Let me write that down. Let z = 7 + 2i. Multiply by i: z * i = (7 + 2i) * i. Let me compute that. Distribute the multiplication: 7*i + 2i*i. Which is 7i + 2i². Since i² is -1, that becomes 7i + 2*(-1) = 7i - 2. Which is -2 + 7i. So the resulting complex number is -2 + 7i.

Wait, let me check again. If the original point is (7, 2), rotating 90 degrees CCW should take it to (-2, 7). Which is indeed -2 + 7i. So that seems correct.

But let me think if there's another way to approach this. Maybe using rotation matrices? Because complex numbers can be represented as vectors in 2D space. A rotation matrix for 90 degrees CCW is [ [0, -1], [1, 0] ]. So if I have a vector (x, y), multiplying by this matrix gives (-y, x). So for (7, 2), that would be (-2, 7), which is the same as -2 + 7i. Yep, same result.

So both methods give me the same answer. That gives me more confidence.

Alternatively, maybe I can think about the modulus and argument. The original complex number 7 + 2i has a modulus of sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53). The argument is arctangent(2/7). When we rotate it by 90 degrees, the modulus remains the same, and the argument increases by π/2. So the new complex number would have modulus sqrt(53) and argument arctan(2/7) + π/2.

But how do I convert that back to rectangular form? Let me see. Let me compute cos(arctan(2/7) + π/2) and sin(arctan(2/7) + π/2).

Alternatively, maybe using angle addition formulas. Let me denote θ = arctan(2/7). Then cos(θ + π/2) = cosθ cos(π/2) - sinθ sin(π/2) = cosθ * 0 - sinθ * 1 = -sinθ. Similarly, sin(θ + π/2) = sinθ cos(π/2) + cosθ sin(π/2) = sinθ * 0 + cosθ * 1 = cosθ.

Therefore, the new complex number is sqrt(53) * [ -sinθ + i cosθ ].

But what is sinθ and cosθ? Since θ is arctan(2/7), we can imagine a right triangle where the opposite side is 2 and adjacent is 7, so hypotenuse is sqrt(7² + 2²) = sqrt(53). Therefore, sinθ = 2 / sqrt(53) and cosθ = 7 / sqrt(53).

Therefore, substituting back, the new complex number is sqrt(53) * [ - (2 / sqrt(53)) + i (7 / sqrt(53)) ] = sqrt(53) * (-2 / sqrt(53)) + sqrt(53) * i * 7 / sqrt(53) = -2 + 7i.

Same result! So that's consistent.

Therefore, regardless of the method I use—multiplying by i, using rotation matrices, or using modulus and argument—the result is -2 + 7i.

Wait, but just to make sure I didn't make any calculation errors. Let me check the multiplication again.

Original number: 7 + 2i. Multiply by i:

(7)(i) + (2i)(i) = 7i + 2i².

Since i² = -1, that's 7i + 2(-1) = 7i - 2.

Which is -2 + 7i. Yep, that's correct.

Alternatively, if I think of the complex plane, the original point is at (7, 2). Rotating 90 degrees CCW swaps the coordinates and changes the sign of the first coordinate. Wait, no. Wait, rotating (x, y) 90 degrees CCW gives (-y, x). So original (7, 2) becomes (-2, 7). Which is indeed -2 + 7i.

So all methods confirm that.

Therefore, I think the answer is definitely -2 + 7i.

But just to be thorough, let me think if there's any chance that the rotation is clockwise instead of counter-clockwise? The problem says counter-clockwise, so that's 90 degrees. If it were clockwise, it would be -90 degrees, which is equivalent to multiplying by e^(-iπ/2) = -i. Then multiplying 7 + 2i by -i would give -7i - 2i² = -7i + 2, which is 2 - 7i. But since the problem says counter-clockwise, that's not the case here.

Alternatively, maybe I confused the direction? Let me check. If I have a point (1, 0), rotating it 90 degrees CCW would take it to (0, 1). If I multiply 1 + 0i by i, I get i, which is (0, 1). Correct. If I rotate (0, 1) 90 degrees CCW, it would go to (-1, 0), which is -1 + 0i. Multiplying i by i gives i² = -1, which is correct. So multiplying by i does indeed rotate CCW.

Therefore, the answer is -2 + 7i.

I think that's solid. All methods converge to the same answer, so I don't see any mistakes here.

**Final Answer**
The resulting complex number is \boxed{-2 + 7i}.

To determine the result of a $90^\circ$ counter-clockwise rotation around the origin applied to the complex number $7 + 2i$, we can use the geometric interpretation of complex numbers in the plane.

---

### Step 1: Understanding the Rotation

In the complex plane, a complex number $z = a + bi$ can be represented as the point $(a, b)$. A rotation of a complex number by an angle $\theta$ counter-clockwise around the origin is achieved by multiplying the complex number by $e^{i\theta}$, or equivalently, by the complex number $\cos\theta + i\sin\theta$.

For a $90^\circ$ rotation, we have:
$$
\theta = 90^\circ = \frac{\pi}{2} \quad \text{radians}
$$
$$
e^{i\pi/2} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = 0 + i = i
$$

So, multiplying a complex number by $i$ rotates it $90^\circ$ counter-clockwise.

---

### Step 2: Apply the Rotation

Let $z = 7 + 2i$. Multiply by $i$:
$$
z \cdot i = (7 + 2i) \cdot i = 7i + 2i^2
$$
Recall that $i^2 = -1$, so:
$$
z \cdot i = 7i + 2(-1) = 7i - 2 = -2 + 7i
$$

---

### Step 3: Interpret the Result

The resulting complex number is $-2 + 7i$, which corresponds to the point $(-2, 7)$ in the complex plane. This is indeed a $90^\circ$ counter-clockwise rotation of the original point $(7, 2)$, as expected.

---

### Final Answer

$$
\boxed{-2 + 7i}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what happens when I rotate the complex number 7 + 2i by 90 degrees counter-clockwise around the origin. Hmm, complex numbers and rotations... I remember that rotations in the complex plane can be done using multiplication by another complex number. Let me recall...

I think a rotation by an angle θ is achieved by multiplying the complex number by e^(iθ). Since we're dealing with a 90-degree rotation, θ is 90 degrees. But wait, in radians, that's π/2. So, e^(iπ/2) is... what's that? Euler's formula says e^(iθ) = cosθ + i sinθ. So cos(π/2) is 0 and sin(π/2) is 1. Therefore, e^(iπ/2) is 0 + i*1 = i.

So, multiplying the original complex number by i should give me the rotated one? Let me check. If I have a complex number z = a + bi, multiplying by i gives z*i = (a + bi)*i = ai + bi^2 = ai - b, since i^2 is -1. So that's -b + ai. Which is like rotating the point (a, b) 90 degrees counter-clockwise. Let me visualize that. If I have a point (a, b), rotating it 90 degrees CCW would take it to (-b, a). Yes, that matches with the result of multiplying by i. So, for example, if I have 1 + 0i, multiplying by i gives 0 + 1i, which is (0,1), which is indeed a 90-degree rotation from (1,0). So that seems right.

Therefore, applying this to 7 + 2i. Let me write that down. Let z = 7 + 2i. Multiply by i: z * i = (7 + 2i) * i. Let me compute that. Distribute the multiplication: 7*i + 2i*i. Which is 7i + 2i². Since i² is -1, that becomes 7i + 2*(-1) = 7i - 2. Which is -2 + 7i. So the resulting complex number is -2 + 7i.

Wait, let me check again. If the original point is (7, 2), rotating 90 degrees CCW should take it to (-2, 7). Which is indeed -2 + 7i. So that seems correct.

But let me think if there's another way to approach this. Maybe using rotation matrices? Because complex numbers can be represented as vectors in 2D space. A rotation matrix for 90 degrees CCW is [ [0, -1], [1, 0] ]. So if I have a vector (x, y), multiplying by this matrix gives (-y, x). So for (7, 2), that would be (-2, 7), which is the same as -2 + 7i. Yep, same result.

So both methods give me the same answer. That gives me more confidence.

Alternatively, maybe I can think about the modulus and argument. The original complex number 7 + 2i has a modulus of sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53). The argument is arctangent(2/7). When we rotate it by 90 degrees, the modulus remains the same, and the argument increases by π/2. So the new complex number would have modulus sqrt(53) and argument arctan(2/7) + π/2.

But how do I convert that back to rectangular form? Let me see. Let me compute cos(arctan(2/7) + π/2) and sin(arctan(2/7) + π/2).

Alternatively, maybe using angle addition formulas. Let me denote θ = arctan(2/7). Then cos(θ + π/2) = cosθ cos(π/2) - sinθ sin(π/2) = cosθ * 0 - sinθ * 1 = -sinθ. Similarly, sin(θ + π/2) = sinθ cos(π/2) + cosθ sin(π/2) = sinθ * 0 + cosθ * 1 = cosθ.

Therefore, the new complex number is sqrt(53) * [ -sinθ + i cosθ ].

But what is sinθ and cosθ? Since θ is arctan(2/7), we can imagine a right triangle where the opposite side is 2 and adjacent is 7, so hypotenuse is sqrt(7² + 2²) = sqrt(53). Therefore, sinθ = 2 / sqrt(53) and cosθ = 7 / sqrt(53).

Therefore, substituting back, the new complex number is sqrt(53) * [ - (2 / sqrt(53)) + i (7 / sqrt(53)) ] = sqrt(53) * (-2 / sqrt(53)) + sqrt(53) * i * 7 / sqrt(53) = -2 + 7i.

Same result! So that's consistent.

Therefore, regardless of the method I use—multiplying by i, using rotation matrices, or using modulus and argument—the result is -2 + 7i.

Wait, but just to make sure I didn't make any calculation errors. Let me check the multiplication again.

Original number: 7 + 2i. Multiply by i:

(7)(i) + (2i)(i) = 7i + 2i².

Since i² = -1, that's 7i + 2(-1) = 7i - 2.

Which is -2 + 7i. Yep, that's correct.

Alternatively, if I think of the complex plane, the original point is at (7, 2). Rotating 90 degrees CCW swaps the coordinates and changes the sign of the first coordinate. Wait, no. Wait, rotating (x, y) 90 degrees CCW gives (-y, x). So original (7, 2) becomes (-2, 7). Which is indeed -2 + 7i.

So all methods confirm that.

Therefore, I think the answer is definitely -2 + 7i.

But just to be thorough, let me think if there's any chance that the rotation is clockwise instead of counter-clockwise? The problem says counter-clockwise, so that's 90 degrees. If it were clockwise, it would be -90 degrees, which is equivalent to multiplying by e^(-iπ/2) = -i. Then multiplying 7 + 2i by -i would give -7i - 2i² = -7i + 2, which is 2 - 7i. But since the problem says counter-clockwise, that's not the case here.

Alternatively, maybe I confused the direction? Let me check. If I have a point (1, 0), rotating it 90 degrees CCW would take it to (0, 1). If I multiply 1 + 0i by i, I get i, which is (0, 1). Correct. If I rotate (0, 1) 90 degrees CCW, it would go to (-1, 0), which is -1 + 0i. Multiplying i by i gives i² = -1, which is correct. So multiplying by i does indeed rotate CCW.

Therefore, the answer is -2 + 7i.

I think that's solid. All methods converge to the same answer, so I don't see any mistakes here.

**Final Answer**
The resulting complex number is \boxed{-2 + 7i}.

To determine the result of a $90^\circ$ counter-clockwise rotation around the origin applied to the complex number $7 + 2i$, we can use the geometric interpretation of complex numbers in the plane.

---

### Step 1: Understanding the Rotation

In the complex plane, a complex number $z = a + bi$ can be represented as the point $(a, b)$. A rotation of a complex number by an angle $\theta$ counter-clockwise around the origin is achieved by multiplying the complex number by $e^{i\theta}$, or equivalently, by the complex number $\cos\theta + i\sin\theta$.

For a $90^\circ$ rotation, we have:
$$
\theta = 90^\circ = \frac{\pi}{2} \quad \text{radians}
$$
$$
e^{i\pi/2} = \cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right) = 0 + i = i
$$

So, multiplying a complex number by $i$ rotates it $90^\circ$ counter-clockwise.

---

### Step 2: Apply the Rotation

Let $z = 7 + 2i$. Multiply by $i$:
$$
z \cdot i = (7 + 2i) \cdot i = 7i + 2i^2
$$
Recall that $i^2 = -1$, so:
$$
z \cdot i = 7i + 2(-1) = 7i - 2 = -2 + 7i
$$

---

### Step 3: Interpret the Result

The resulting complex number is $-2 + 7i$, which corresponds to the point $(-2, 7)$ in the complex plane. This is indeed a $90^\circ$ counter-clockwise rotation of the original point $(7, 2)$, as expected.

---

### Final Answer

$$
\boxed{-2 + 7i}
$$

Solution:
A $90^\circ$ rotation around the origin in the counter-clockwise direction corresponds to multiplication by $\operatorname{cis} 90^\circ = i.$

[asy]
unitsize(0.5 cm);

draw((-3,0)--(8,0));
draw((0,-1)--(0,8));
draw((0,0)--(7,2),dashed);
draw((0,0)--(-2,7),dashed);

dot("$7 + 2i$", (7,2), E);
dot("$-2 + 7i$", (-2,7), N);
[/asy]

Thus, the image of $7 + 2i$ is $i(7 + 2i) = \boxed{-2 + 7i}.$