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 would be sqrt(0² + 3²) which is 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 yes, θ is π/2.

But wait, let me make sure I'm not missing something here. Sometimes when converting, especially with arctangent, you have to consider the quadrant. But since x is 0 and y is positive, the point is on the positive y-axis, which is the boundary between the first and second quadrants. But since x is zero, it's not in any quadrant, but rather on the axis.

So in this case, since x is zero and y is positive, θ is π/2. So putting it all together, the polar coordinates should be (3, π/2).

But let me check if there's another way to represent this. For example, sometimes angles can be represented with different values by adding multiples of 2π, but the question specifies that θ should be between 0 and 2π, so π/2 is already in that interval.

Alternatively, if I use the arctangent function, even though y/x is undefined, maybe I can use the arctangent of y over x with some adjustments. Let me think. If I take the limit as x approaches 0 from the positive side, arctan(y/x) would approach arctan(infinity) which is π/2. Similarly, if x approaches 0 from the negative side, arctan(y/x) would approach arctan(-infinity), which is -π/2, but since we are in the positive y-axis, we need to adjust that.

But since x is exactly 0 here, not approaching it, maybe the standard way is to recognize that it's on the y-axis. So θ is π/2.

Alternatively, maybe I can use the atan2 function, which takes into account both x and y. The atan2(y, x) function would give me the correct angle. Let me recall that. If I use atan2(3, 0), what does that give me?

I think atan2(y, x) returns the angle θ such that x = r cos θ and y = r sin θ. So if x is 0 and y is positive, then θ is π/2. So yes, that would be the case.

Therefore, θ is π/2.

So putting it all together, the polar coordinates are (r, θ) = (3, π/2).

Wait, let me just check if there's any other possible value for θ. For example, if I add 2π to π/2, that would be 5π/2, but that's more than 2π, so it's not in the required range. If I subtract 2π, that would be negative, which is also not in the required range. So π/2 is the only valid angle in [0, 2π).

Therefore, I think the answer is (3, π/2).

But let me just visualize the point. The original point is (0, 3), which is 3 units up the y-axis. In polar coordinates, that should be a radius of 3, and angle of 90 degrees or π/2 radians. That makes sense.

Alternatively, if I think about the conversion formulas again.

r = sqrt(x² + y²) = sqrt(0 + 9) = 3.

θ = arctan(y/x). But since x is 0, we can't compute arctan(3/0). However, since x is 0 and y is positive, θ is π/2.

Alternatively, if I use the formula θ = arctan(y/x) + π if x < 0, but since x is 0, that formula isn't directly applicable.

But in general, when converting, if x is 0, then θ is π/2 if y is positive, and 3π/2 if y is negative.

So yes, since y is positive here, θ is π/2.

Therefore, I think I can confidently say that the polar coordinates are (3, π/2).

Just to make sure, let me convert it back to rectangular coordinates to check.

If r = 3 and θ = π/2, then x = r cos θ = 3 cos(π/2) = 3 * 0 = 0.

y = r sin θ = 3 sin(π/2) = 3 * 1 = 3.

So that gives back the original point (0, 3). Perfect, that checks out.

Therefore, the answer is (3, π/2).

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

To convert the rectangular coordinates $(0, 3)$ to polar coordinates, 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 $ \frac{y}{x} $ is undefined. This indicates that the point lies on the **y-axis**.

- Since $ y = 3 > 0 $, the point is on the **positive y-axis**.
- The angle between the positive x-axis and the positive y-axis is $ \frac{\pi}{2} $ radians.

Thus, $ \theta = \frac{\pi}{2} $.

---

### Step 3: Verify the Result

To ensure correctness, we can convert back from polar to rectangular coordinates:

- $ x = r \cos(\theta) = 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 $
- $ y = r \sin(\theta) = 3 \sin\left(\frac{\pi}{2}\right) = 3 \cdot 1 = 3 $

This gives us the original point $(0, 3)$, confirming our conversion is accurate.

---

### 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 would be sqrt(0² + 3²) which is 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 yes, θ is π/2.

But wait, let me make sure I'm not missing something here. Sometimes when converting, especially with arctangent, you have to consider the quadrant. But since x is 0 and y is positive, the point is on the positive y-axis, which is the boundary between the first and second quadrants. But since x is zero, it's not in any quadrant, but rather on the axis.

So in this case, since x is zero and y is positive, θ is π/2. So putting it all together, the polar coordinates should be (3, π/2).

But let me check if there's another way to represent this. For example, sometimes angles can be represented with different values by adding multiples of 2π, but the question specifies that θ should be between 0 and 2π, so π/2 is already in that interval.

Alternatively, if I use the arctangent function, even though y/x is undefined, maybe I can use the arctangent of y over x with some adjustments. Let me think. If I take the limit as x approaches 0 from the positive side, arctan(y/x) would approach arctan(infinity) which is π/2. Similarly, if x approaches 0 from the negative side, arctan(y/x) would approach arctan(-infinity), which is -π/2, but since we are in the positive y-axis, we need to adjust that.

But since x is exactly 0 here, not approaching it, maybe the standard way is to recognize that it's on the y-axis. So θ is π/2.

Alternatively, maybe I can use the atan2 function, which takes into account both x and y. The atan2(y, x) function would give me the correct angle. Let me recall that. If I use atan2(3, 0), what does that give me?

I think atan2(y, x) returns the angle θ such that x = r cos θ and y = r sin θ. So if x is 0 and y is positive, then θ is π/2. So yes, that would be the case.

Therefore, θ is π/2.

So putting it all together, the polar coordinates are (r, θ) = (3, π/2).

Wait, let me just check if there's any other possible value for θ. For example, if I add 2π to π/2, that would be 5π/2, but that's more than 2π, so it's not in the required range. If I subtract 2π, that would be negative, which is also not in the required range. So π/2 is the only valid angle in [0, 2π).

Therefore, I think the answer is (3, π/2).

But let me just visualize the point. The original point is (0, 3), which is 3 units up the y-axis. In polar coordinates, that should be a radius of 3, and angle of 90 degrees or π/2 radians. That makes sense.

Alternatively, if I think about the conversion formulas again.

r = sqrt(x² + y²) = sqrt(0 + 9) = 3.

θ = arctan(y/x). But since x is 0, we can't compute arctan(3/0). However, since x is 0 and y is positive, θ is π/2.

Alternatively, if I use the formula θ = arctan(y/x) + π if x < 0, but since x is 0, that formula isn't directly applicable.

But in general, when converting, if x is 0, then θ is π/2 if y is positive, and 3π/2 if y is negative.

So yes, since y is positive here, θ is π/2.

Therefore, I think I can confidently say that the polar coordinates are (3, π/2).

Just to make sure, let me convert it back to rectangular coordinates to check.

If r = 3 and θ = π/2, then x = r cos θ = 3 cos(π/2) = 3 * 0 = 0.

y = r sin θ = 3 sin(π/2) = 3 * 1 = 3.

So that gives back the original point (0, 3). Perfect, that checks out.

Therefore, the answer is (3, π/2).

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

To convert the rectangular coordinates $(0, 3)$ to polar coordinates, 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 $ \frac{y}{x} $ is undefined. This indicates that the point lies on the **y-axis**.

- Since $ y = 3 > 0 $, the point is on the **positive y-axis**.
- The angle between the positive x-axis and the positive y-axis is $ \frac{\pi}{2} $ radians.

Thus, $ \theta = \frac{\pi}{2} $.

---

### Step 3: Verify the Result

To ensure correctness, we can convert back from polar to rectangular coordinates:

- $ x = r \cos(\theta) = 3 \cos\left(\frac{\pi}{2}\right) = 3 \cdot 0 = 0 $
- $ y = r \sin(\theta) = 3 \sin\left(\frac{\pi}{2}\right) = 3 \cdot 1 = 3 $

This gives us the original point $(0, 3)$, confirming our conversion is accurate.

---

### 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 point on the graph, maybe I need to consider the distance at a specific time? Hmm, maybe I need to check the coordinates of each student's dot.

Looking at the Asymptote code, there are five students: Evelyn, Briana, Carla, Debra, and Angela. Each has a dot at specific coordinates. Let me list them out:

- 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)

Wait, but in a distance-time graph, the x-axis is time and the y-axis is distance. So each student's position is at a certain time (x-coordinate) and a certain distance (y-coordinate). But if we're talking about average speed, we need to know the total distance covered over the total time. However, if each student is just a single point, does that mean they started at time 0 and ended at that time? Or is the graph showing their position at a specific time?

Wait, maybe the graph is showing the distance covered at different times. So each student's dot is at a particular time (x-axis) and distance (y-axis). So if we want average speed, we need to calculate the distance divided by time for each student. But if they are only at one point, maybe they are at that distance at that time, but we don't know their starting point? Wait, but the graph is labeled with O at (0,0), which is the origin. Maybe each student started at O (0,0) and then ran to their respective points? If that's the case, then their average speed would be their distance divided by time. So for each student, the y-coordinate is the distance, and the x-coordinate is the time. Therefore, average speed is distance (y) divided by time (x). So we can calculate that for each student and compare.

Let me check that assumption. If the graph is a distance-time graph, then the position of each student at a certain time is given by their coordinates. If they started at (0,0), then their average speed would be (distance at time t) / t. So yes, that seems to be the case here. So let me calculate that for each student.

Starting with Evelyn: (1.25, 4.5). So time is 1.25 units, distance is 4.5 units. So average speed is 4.5 / 1.25. Let me compute that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 4.5 minus 3.75 is 0.75. So 0.75 / 1.25 is 0.6. So total is 3.6. So 3.6 units per time unit.

Briana: (2.5, 2.2). So average speed is 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.

Carla: (4.25, 5.2). So 5.2 divided by 4.25. Let me compute that. 4.25 times 1 is 4.25, subtract that from 5.2, we get 0.95. Then, 0.95 divided by 4.25. Let me convert to fractions. 5.2 / 4.25. Multiply numerator and denominator by 100 to eliminate decimals: 520 / 425. Simplify. Divide numerator and denominator by 5: 104 / 85. Let me divide 104 by 85. 85 goes into 104 once, with remainder 19. So approximately 1.2235... So approximately 1.22 units per time unit.

Debra: (5.6, 2.8). So average speed is 2.8 / 5.6. That's 0.5 units per time unit.

Angela: (6.8, 1.4). So 1.4 / 6.8. Let me compute that. 1.4 divided by 6.8. Multiply numerator and denominator by 10: 14 / 68. Simplify: 7 / 34 ≈ 0.2058... So approximately 0.206 units per time unit.

So summarizing:

- Evelyn: 3.6
- Briana: 0.88
- Carla: ~1.22
- Debra: 0.5
- Angela: ~0.206

So the greatest average speed is Evelyn with 3.6. Therefore, Evelyn has the greatest average speed.

Wait, but let me double-check my calculations to make sure I didn't make any errors.

Starting with Evelyn: 4.5 divided by 1.25. Let me do this another way. 1.25 is 5/4. So dividing by 5/4 is multiplying by 4/5. So 4.5 * 4 / 5. 4.5 * 4 is 18, divided by 5 is 3.6. Correct.

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

Carla: 5.2 / 4.25. Let me convert 4.25 to a fraction. 4.25 is 17/4. So 5.2 divided by 17/4 is 5.2 * 4 / 17. 5.2 * 4 is 20.8. 20.8 / 17. Let me compute that. 17*1.2 = 20.4. So 20.8 - 20.4 = 0.4. So 1.2 + 0.4/17 ≈ 1.2 + 0.0235 ≈ 1.2235. So approximately 1.22. Correct.

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

Angela: 1.4 / 6.8. Let me do decimal division. 1.4 divided by 6.8. Since 1.4 is less than 6.8, it's 0.2... Let me compute 1.4 / 6.8. Multiply numerator and denominator by 10: 14 / 68. Divide numerator and denominator by 2: 7 / 34. Let me divide 7 by 34. 34 goes into 7 zero times. Add decimal: 70 divided by 34 is 2 (34*2=68), remainder 2. Bring down 0: 20. 34 goes into 20 zero times. Bring down next 0: 200. 34*5=170, 34*6=204. So 5. So 0.205... So approximately 0.205. Correct.

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

But wait, let me check if there's another interpretation. Maybe the graph is showing the distance covered over time, so each student's path is represented by a point, but maybe they are not starting from the origin? Wait, the graph is labeled with O at (0,0), which is the origin. The Asymptote code draws the axes from (0,0) to (8,0) and (0,0) to (0,6). The dots are plotted at various positions. The labels for the students are placed near their respective points. So if each student is at a specific time and distance, then if they started at O, then their average speed is distance over time. But if they didn't start at O, then we wouldn't have enough information. However, since the graph is a distance-time graph, and the origin is labeled as O, it's reasonable to assume that each student started at O (0,0) and then ran to their respective points. Therefore, the average speed is distance divided by time. So Evelyn's average speed is 4.5 / 1.25 = 3.6, which is the highest. Therefore, Evelyn is the answer.

Alternatively, if the graph was showing something else, like position vs. time for each student, but with multiple points, but in this case, each student is only one point. So maybe the question is about instantaneous speed? But the question says average speed. So average speed would be total distance over total time. If they started at O, then yes, it's the distance at that time divided by time. So I think my reasoning is correct.

Wait, but another thought: if the graph is a distance-time graph, then the slope of the line from the origin to the point would represent the average speed. So for each student, if we draw a line from (0,0) to their point, the slope is (y - 0)/(x - 0) = y/x, which is exactly what I calculated. So Evelyn's slope is 4.5 / 1.25 = 3.6, which 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: the y-axis is distance, and the x-axis is time. So the coordinates are (time, distance). So for example, Evelyn is at time 1.25, distance 4.5. So yes, average speed is distance over time. So my calculations are 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**. In such graphs, the **x-axis represents time** and the **y-axis represents distance**. Each student's position is marked by a dot at a specific (time, distance) coordinate.

---

### Step 1: Understand the Formula for Average Speed

Average speed is defined as:

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

In this context, since each student is represented by a single point on the graph, we assume they started at the origin (0,0) and ran to their respective point. Therefore, the **average speed** for each student is:

$$
\text{Average Speed} = \frac{\text{Distance (y-coordinate)}}{\text{Time (x-coordinate)}}
$$

---

### Step 2: Calculate 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)
  $$
  \text{Speed} = \frac{5.2}{4.25} \approx 1.22
  $$

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

- **Angela**: (6.8, 1.4)
  $$
  \text{Speed} = \frac{1.4}{6.8} \approx 0.206
  $$

---

### Step 3: Compare the Results

Now, comparing the average speeds:

- Evelyn: **3.6**
- Briana: **0.88**
- Carla: **1.22**
- Debra: **0.5**
- Angela: **0.206**

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 point on the graph, maybe I need to consider the distance at a specific time? Hmm, maybe I need to check the coordinates of each student's dot.

Looking at the Asymptote code, there are five students: Evelyn, Briana, Carla, Debra, and Angela. Each has a dot at specific coordinates. Let me list them out:

- 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)

Wait, but in a distance-time graph, the x-axis is time and the y-axis is distance. So each student's position is at a certain time (x-coordinate) and a certain distance (y-coordinate). But if we're talking about average speed, we need to know the total distance covered over the total time. However, if each student is just a single point, does that mean they started at time 0 and ended at that time? Or is the graph showing their position at a specific time?

Wait, maybe the graph is showing the distance covered at different times. So each student's dot is at a particular time (x-axis) and distance (y-axis). So if we want average speed, we need to calculate the distance divided by time for each student. But if they are only at one point, maybe they are at that distance at that time, but we don't know their starting point? Wait, but the graph is labeled with O at (0,0), which is the origin. Maybe each student started at O (0,0) and then ran to their respective points? If that's the case, then their average speed would be their distance divided by time. So for each student, the y-coordinate is the distance, and the x-coordinate is the time. Therefore, average speed is distance (y) divided by time (x). So we can calculate that for each student and compare.

Let me check that assumption. If the graph is a distance-time graph, then the position of each student at a certain time is given by their coordinates. If they started at (0,0), then their average speed would be (distance at time t) / t. So yes, that seems to be the case here. So let me calculate that for each student.

Starting with Evelyn: (1.25, 4.5). So time is 1.25 units, distance is 4.5 units. So average speed is 4.5 / 1.25. Let me compute that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 4.5 minus 3.75 is 0.75. So 0.75 / 1.25 is 0.6. So total is 3.6. So 3.6 units per time unit.

Briana: (2.5, 2.2). So average speed is 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.

Carla: (4.25, 5.2). So 5.2 divided by 4.25. Let me compute that. 4.25 times 1 is 4.25, subtract that from 5.2, we get 0.95. Then, 0.95 divided by 4.25. Let me convert to fractions. 5.2 / 4.25. Multiply numerator and denominator by 100 to eliminate decimals: 520 / 425. Simplify. Divide numerator and denominator by 5: 104 / 85. Let me divide 104 by 85. 85 goes into 104 once, with remainder 19. So approximately 1.2235... So approximately 1.22 units per time unit.

Debra: (5.6, 2.8). So average speed is 2.8 / 5.6. That's 0.5 units per time unit.

Angela: (6.8, 1.4). So 1.4 / 6.8. Let me compute that. 1.4 divided by 6.8. Multiply numerator and denominator by 10: 14 / 68. Simplify: 7 / 34 ≈ 0.2058... So approximately 0.206 units per time unit.

So summarizing:

- Evelyn: 3.6
- Briana: 0.88
- Carla: ~1.22
- Debra: 0.5
- Angela: ~0.206

So the greatest average speed is Evelyn with 3.6. Therefore, Evelyn has the greatest average speed.

Wait, but let me double-check my calculations to make sure I didn't make any errors.

Starting with Evelyn: 4.5 divided by 1.25. Let me do this another way. 1.25 is 5/4. So dividing by 5/4 is multiplying by 4/5. So 4.5 * 4 / 5. 4.5 * 4 is 18, divided by 5 is 3.6. Correct.

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

Carla: 5.2 / 4.25. Let me convert 4.25 to a fraction. 4.25 is 17/4. So 5.2 divided by 17/4 is 5.2 * 4 / 17. 5.2 * 4 is 20.8. 20.8 / 17. Let me compute that. 17*1.2 = 20.4. So 20.8 - 20.4 = 0.4. So 1.2 + 0.4/17 ≈ 1.2 + 0.0235 ≈ 1.2235. So approximately 1.22. Correct.

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

Angela: 1.4 / 6.8. Let me do decimal division. 1.4 divided by 6.8. Since 1.4 is less than 6.8, it's 0.2... Let me compute 1.4 / 6.8. Multiply numerator and denominator by 10: 14 / 68. Divide numerator and denominator by 2: 7 / 34. Let me divide 7 by 34. 34 goes into 7 zero times. Add decimal: 70 divided by 34 is 2 (34*2=68), remainder 2. Bring down 0: 20. 34 goes into 20 zero times. Bring down next 0: 200. 34*5=170, 34*6=204. So 5. So 0.205... So approximately 0.205. Correct.

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

But wait, let me check if there's another interpretation. Maybe the graph is showing the distance covered over time, so each student's path is represented by a point, but maybe they are not starting from the origin? Wait, the graph is labeled with O at (0,0), which is the origin. The Asymptote code draws the axes from (0,0) to (8,0) and (0,0) to (0,6). The dots are plotted at various positions. The labels for the students are placed near their respective points. So if each student is at a specific time and distance, then if they started at O, then their average speed is distance over time. But if they didn't start at O, then we wouldn't have enough information. However, since the graph is a distance-time graph, and the origin is labeled as O, it's reasonable to assume that each student started at O (0,0) and then ran to their respective points. Therefore, the average speed is distance divided by time. So Evelyn's average speed is 4.5 / 1.25 = 3.6, which is the highest. Therefore, Evelyn is the answer.

Alternatively, if the graph was showing something else, like position vs. time for each student, but with multiple points, but in this case, each student is only one point. So maybe the question is about instantaneous speed? But the question says average speed. So average speed would be total distance over total time. If they started at O, then yes, it's the distance at that time divided by time. So I think my reasoning is correct.

Wait, but another thought: if the graph is a distance-time graph, then the slope of the line from the origin to the point would represent the average speed. So for each student, if we draw a line from (0,0) to their point, the slope is (y - 0)/(x - 0) = y/x, which is exactly what I calculated. So Evelyn's slope is 4.5 / 1.25 = 3.6, which 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: the y-axis is distance, and the x-axis is time. So the coordinates are (time, distance). So for example, Evelyn is at time 1.25, distance 4.5. So yes, average speed is distance over time. So my calculations are 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**. In such graphs, the **x-axis represents time** and the **y-axis represents distance**. Each student's position is marked by a dot at a specific (time, distance) coordinate.

---

### Step 1: Understand the Formula for Average Speed

Average speed is defined as:

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

In this context, since each student is represented by a single point on the graph, we assume they started at the origin (0,0) and ran to their respective point. Therefore, the **average speed** for each student is:

$$
\text{Average Speed} = \frac{\text{Distance (y-coordinate)}}{\text{Time (x-coordinate)}}
$$

---

### Step 2: Calculate 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)
  $$
  \text{Speed} = \frac{5.2}{4.25} \approx 1.22
  $$

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

- **Angela**: (6.8, 1.4)
  $$
  \text{Speed} = \frac{1.4}{6.8} \approx 0.206
  $$

---

### Step 3: Compare the Results

Now, comparing the average speeds:

- Evelyn: **3.6**
- Briana: **0.88**
- Carla: **1.22**
- Debra: **0.5**
- Angela: **0.206**

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? Maybe by drawing lines from the center to each of the six vertices? That would create six congruent triangles, right?

Yes, if you connect all the vertices to the center, you divide the hexagon into six equilateral triangles. 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 since it's a regular hexagon, all the sides are equal, and all the radii are equal too. So, if those triangles are equilateral, then all their sides must be equal. That means the length of each side of the hexagon is equal to the radius of the circumscribed circle around the hexagon.

But wait, if the triangles are equilateral, then each triangle has three sides of equal length. So, if the perimeter of one triangle is 21 inches, then each side of the triangle is 21 divided by 3. Let me calculate that.

Perimeter of triangle = 21 inches. Since it's an equilateral triangle, all sides are equal. So each side length is 21 / 3 = 7 inches. Therefore, each side of the triangle is 7 inches. But since the triangles are formed by connecting the center to the vertices of the hexagon, each side of the triangle is equal to the side of the hexagon. Wait, is that correct?

Hold on, let me think again. If you have a regular hexagon, and you connect the center to each vertex, each of those triangles is an equilateral triangle. So, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. But in a regular hexagon, the side length is equal to the radius. So, if the triangles are equilateral, then the radius is equal to the side length of the hexagon. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, if the perimeter of the triangle is 21 inches, then each side of the triangle is 7 inches, which would mean that each side of the hexagon is 7 inches. Therefore, the perimeter of the hexagon would be 6 times 7 inches, which is 42 inches. Is that right?

Wait, but let me confirm. Let me recall that in a regular hexagon, the side length is equal to the radius. So, if you divide the hexagon into six equilateral triangles, each triangle has sides equal to the radius. Therefore, the side length of the hexagon is equal to the radius. Therefore, if each triangle has a perimeter of 21 inches, then each side of the triangle is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches. That seems straightforward.

But let me check if there's another way to think about this. Maybe I'm missing something. Let me consider the properties of a regular hexagon. A regular hexagon can be divided into six equilateral triangles by drawing lines from the center to each vertex. Each of those triangles is equilateral. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, if the perimeter of each triangle is 21 inches, then each side is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches. So, that seems consistent.

Alternatively, maybe I can think about the relationship between the triangles and the hexagon. Each triangle has three sides. But in the hexagon, each side is shared by two triangles? Wait, no. Wait, when you divide the hexagon into six triangles by connecting the center to each vertex, each triangle has one side that is a side of the hexagon, and the other two sides are radii. Therefore, each triangle's side that is a side of the hexagon is unique to that triangle. Therefore, each triangle's side length is equal to the side length of the hexagon. Therefore, if each triangle has a perimeter of 21 inches, then each side of the triangle is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches.

But let me think again. Suppose the triangle has sides of length 's'. Then the perimeter is 3s = 21, so s=7. Therefore, each triangle has sides of 7 inches. Since each triangle's side that is part of the hexagon is 7 inches, then the hexagon has sides of 7 inches. Therefore, the perimeter is 6*7=42 inches. So, that seems straightforward.

But maybe I need to check if the triangles are actually equilateral. Wait, the problem says that a regular hexagon can be divided into six equilateral triangles. So, that's given. Therefore, we don't need to verify that. So, given that, then the logic holds.

Alternatively, maybe there's confusion between the triangles formed by connecting the center to the vertices versus other triangles. For example, if you divide the hexagon into different triangles, but the problem says that it can be divided into six equilateral triangles, so that must be the case. Therefore, each triangle is equilateral with side length equal to the side length of the hexagon.

Therefore, the perimeter of each triangle is 3 times the side length. Therefore, 3s = 21, so s=7. Therefore, the perimeter of the hexagon is 6s = 6*7=42 inches. Therefore, the answer is 42 inches.

But let me think again. Maybe there's a different way to divide the hexagon into six equilateral triangles? For example, if you divide it into smaller triangles? Wait, but the problem says "a regular hexagon can be divided into six equilateral triangles." So, the standard way is connecting the center to the vertices, which gives six equilateral triangles. So, that's the division they are referring to.

Alternatively, if you divide the hexagon into smaller triangles, but then the number of triangles would be more than six. For example, if you divide each of the six equilateral triangles into smaller triangles, but the problem says six triangles. Therefore, the division is into six equilateral triangles, which is done by connecting the center to the vertices. Therefore, each triangle has two sides that are radii and one side that is a side of the hexagon. Since the hexagon is regular, all sides are equal, and all radii are equal. Therefore, each triangle is equilateral. Therefore, the side length of the triangle is equal to the side length of the hexagon. Therefore, perimeter of triangle is 3s = 21, so s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42.

Alternatively, maybe the problem is trying to trick me? Let me check again. Suppose that the triangles are not the ones connecting the center. Suppose that the hexagon is divided into six equilateral triangles in some other way. But how?

For example, if you divide the hexagon into six equilateral triangles by drawing lines from one vertex to the opposite sides? But that might not result in equilateral triangles. Let me think. If you take a regular hexagon and divide it into six equilateral triangles, the only way is by connecting the center to each vertex. Because otherwise, if you divide it in another way, the triangles might not be equilateral. For example, if you divide it into triangles by drawing lines from one vertex to all non-adjacent vertices, but that would create different shapes.

Alternatively, maybe the problem is referring to dividing the hexagon into six equilateral triangles by some other method. But given that the problem says "a regular hexagon can be divided into six equilateral triangles," and given that the standard division is connecting the center to the vertices, which gives six equilateral triangles, then that must be the case.

Therefore, I think my reasoning is correct. Each triangle has sides equal to the side length of the hexagon, so each triangle's perimeter is 3 times the side length. Therefore, 3s = 21, so s=7. Therefore, the perimeter of the hexagon is 6*7=42 inches.

But let me check with actual measurements. Let me recall that in a regular hexagon, the side length is equal to the radius. So, if you have a regular hexagon with side length 's', then the distance from the center to any vertex (the radius) is also 's'. Therefore, when you connect the center to the vertices, each triangle is equilateral with sides of length 's'. Therefore, perimeter of each triangle is 3s. Therefore, given that 3s=21, s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42.

Alternatively, maybe I can think of the regular hexagon as being composed of six equilateral triangles, each with side length 's', so the area of the hexagon would be 6 times the area of one triangle. But the problem isn't asking about area, it's asking about perimeter. But maybe this is just extra.

Alternatively, maybe I can think of the perimeter of the hexagon. Since each side of the hexagon is equal to the side of the triangle, which is 7 inches, so 6 sides would be 42 inches. Therefore, answer is 42.

But let me check if there's a possibility that the triangles have different side lengths. For example, if the triangles are not the ones with the center. Suppose that the hexagon is divided into six equilateral triangles in a different way. For example, if you divide it into triangles by drawing lines from one vertex to the opposite sides. But in that case, the triangles would not be equilateral. Let me try to imagine.

Take a regular hexagon. Let me label the vertices A, B, C, D, E, F. If I connect vertex A to the midpoint of the opposite side, but that would not create an equilateral triangle. Alternatively, if I divide the hexagon into six equilateral triangles, maybe by some other method. But I don't think that's possible. Because the regular hexagon has angles of 120 degrees. If you try to divide it into equilateral triangles, which have 60-degree angles, you need to have specific divisions.

But the standard division is connecting the center to the vertices, which gives six equilateral triangles. Therefore, that must be the division they are referring to. Therefore, the answer is 42 inches.

Alternatively, maybe the problem is referring to the triangles formed by splitting the hexagon into smaller triangles. For example, if you divide each of the six equilateral triangles into smaller triangles, but then the number of triangles would be more than six. However, the problem says six triangles. Therefore, the only way is the standard division.

Therefore, I think my answer is correct. The perimeter of the hexagon is 42 inches.

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

To determine the perimeter of a regular hexagon that is divided into six equilateral triangles, we begin by analyzing the geometric relationship between the hexagon and the triangles.

---

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

A **regular hexagon** has six equal sides and six equal angles. When you connect the **center** of the hexagon to each of its **six vertices**, the hexagon is divided into **six congruent equilateral triangles**. Each triangle has:

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

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

---

### Step 2: Use the Given Perimeter of a Triangle

We are told that the **perimeter of one of these equilateral triangles is 21 inches**. Since all sides of an equilateral triangle are equal, we can find the length of one side by dividing the perimeter by 3:

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

This side length is also the **side length of the hexagon**, because each triangle shares one side with the hexagon.

---

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

Now that we know the side length of the hexagon is 7 inches, we can compute its perimeter by multiplying the side length by the number of sides:

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

---

### 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? Maybe by drawing lines from the center to each of the six vertices? That would create six congruent triangles, right?

Yes, if you connect all the vertices to the center, you divide the hexagon into six equilateral triangles. 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 since it's a regular hexagon, all the sides are equal, and all the radii are equal too. So, if those triangles are equilateral, then all their sides must be equal. That means the length of each side of the hexagon is equal to the radius of the circumscribed circle around the hexagon.

But wait, if the triangles are equilateral, then each triangle has three sides of equal length. So, if the perimeter of one triangle is 21 inches, then each side of the triangle is 21 divided by 3. Let me calculate that.

Perimeter of triangle = 21 inches. Since it's an equilateral triangle, all sides are equal. So each side length is 21 / 3 = 7 inches. Therefore, each side of the triangle is 7 inches. But since the triangles are formed by connecting the center to the vertices of the hexagon, each side of the triangle is equal to the side of the hexagon. Wait, is that correct?

Hold on, let me think again. If you have a regular hexagon, and you connect the center to each vertex, each of those triangles is an equilateral triangle. So, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. But in a regular hexagon, the side length is equal to the radius. So, if the triangles are equilateral, then the radius is equal to the side length of the hexagon. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, if the perimeter of the triangle is 21 inches, then each side of the triangle is 7 inches, which would mean that each side of the hexagon is 7 inches. Therefore, the perimeter of the hexagon would be 6 times 7 inches, which is 42 inches. Is that right?

Wait, but let me confirm. Let me recall that in a regular hexagon, the side length is equal to the radius. So, if you divide the hexagon into six equilateral triangles, each triangle has sides equal to the radius. Therefore, the side length of the hexagon is equal to the radius. Therefore, if each triangle has a perimeter of 21 inches, then each side of the triangle is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches. That seems straightforward.

But let me check if there's another way to think about this. Maybe I'm missing something. Let me consider the properties of a regular hexagon. A regular hexagon can be divided into six equilateral triangles by drawing lines from the center to each vertex. Each of those triangles is equilateral. Therefore, each triangle has sides equal to the side length of the hexagon. Therefore, if the perimeter of each triangle is 21 inches, then each side is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches. So, that seems consistent.

Alternatively, maybe I can think about the relationship between the triangles and the hexagon. Each triangle has three sides. But in the hexagon, each side is shared by two triangles? Wait, no. Wait, when you divide the hexagon into six triangles by connecting the center to each vertex, each triangle has one side that is a side of the hexagon, and the other two sides are radii. Therefore, each triangle's side that is a side of the hexagon is unique to that triangle. Therefore, each triangle's side length is equal to the side length of the hexagon. Therefore, if each triangle has a perimeter of 21 inches, then each side of the triangle is 7 inches, so the side length of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6*7=42 inches.

But let me think again. Suppose the triangle has sides of length 's'. Then the perimeter is 3s = 21, so s=7. Therefore, each triangle has sides of 7 inches. Since each triangle's side that is part of the hexagon is 7 inches, then the hexagon has sides of 7 inches. Therefore, the perimeter is 6*7=42 inches. So, that seems straightforward.

But maybe I need to check if the triangles are actually equilateral. Wait, the problem says that a regular hexagon can be divided into six equilateral triangles. So, that's given. Therefore, we don't need to verify that. So, given that, then the logic holds.

Alternatively, maybe there's confusion between the triangles formed by connecting the center to the vertices versus other triangles. For example, if you divide the hexagon into different triangles, but the problem says that it can be divided into six equilateral triangles, so that must be the case. Therefore, each triangle is equilateral with side length equal to the side length of the hexagon.

Therefore, the perimeter of each triangle is 3 times the side length. Therefore, 3s = 21, so s=7. Therefore, the perimeter of the hexagon is 6s = 6*7=42 inches. Therefore, the answer is 42 inches.

But let me think again. Maybe there's a different way to divide the hexagon into six equilateral triangles? For example, if you divide it into smaller triangles? Wait, but the problem says "a regular hexagon can be divided into six equilateral triangles." So, the standard way is connecting the center to the vertices, which gives six equilateral triangles. So, that's the division they are referring to.

Alternatively, if you divide the hexagon into smaller triangles, but then the number of triangles would be more than six. For example, if you divide each of the six equilateral triangles into smaller triangles, but the problem says six triangles. Therefore, the division is into six equilateral triangles, which is done by connecting the center to the vertices. Therefore, each triangle has two sides that are radii and one side that is a side of the hexagon. Since the hexagon is regular, all sides are equal, and all radii are equal. Therefore, each triangle is equilateral. Therefore, the side length of the triangle is equal to the side length of the hexagon. Therefore, perimeter of triangle is 3s = 21, so s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42.

Alternatively, maybe the problem is trying to trick me? Let me check again. Suppose that the triangles are not the ones connecting the center. Suppose that the hexagon is divided into six equilateral triangles in some other way. But how?

For example, if you divide the hexagon into six equilateral triangles by drawing lines from one vertex to the opposite sides? But that might not result in equilateral triangles. Let me think. If you take a regular hexagon and divide it into six equilateral triangles, the only way is by connecting the center to each vertex. Because otherwise, if you divide it in another way, the triangles might not be equilateral. For example, if you divide it into triangles by drawing lines from one vertex to all non-adjacent vertices, but that would create different shapes.

Alternatively, maybe the problem is referring to dividing the hexagon into six equilateral triangles by some other method. But given that the problem says "a regular hexagon can be divided into six equilateral triangles," and given that the standard division is connecting the center to the vertices, which gives six equilateral triangles, then that must be the case.

Therefore, I think my reasoning is correct. Each triangle has sides equal to the side length of the hexagon, so each triangle's perimeter is 3 times the side length. Therefore, 3s = 21, so s=7. Therefore, the perimeter of the hexagon is 6*7=42 inches.

But let me check with actual measurements. Let me recall that in a regular hexagon, the side length is equal to the radius. So, if you have a regular hexagon with side length 's', then the distance from the center to any vertex (the radius) is also 's'. Therefore, when you connect the center to the vertices, each triangle is equilateral with sides of length 's'. Therefore, perimeter of each triangle is 3s. Therefore, given that 3s=21, s=7. Therefore, perimeter of hexagon is 6s=42. Therefore, answer is 42.

Alternatively, maybe I can think of the regular hexagon as being composed of six equilateral triangles, each with side length 's', so the area of the hexagon would be 6 times the area of one triangle. But the problem isn't asking about area, it's asking about perimeter. But maybe this is just extra.

Alternatively, maybe I can think of the perimeter of the hexagon. Since each side of the hexagon is equal to the side of the triangle, which is 7 inches, so 6 sides would be 42 inches. Therefore, answer is 42.

But let me check if there's a possibility that the triangles have different side lengths. For example, if the triangles are not the ones with the center. Suppose that the hexagon is divided into six equilateral triangles in a different way. For example, if you divide it into triangles by drawing lines from one vertex to the opposite sides. But in that case, the triangles would not be equilateral. Let me try to imagine.

Take a regular hexagon. Let me label the vertices A, B, C, D, E, F. If I connect vertex A to the midpoint of the opposite side, but that would not create an equilateral triangle. Alternatively, if I divide the hexagon into six equilateral triangles, maybe by some other method. But I don't think that's possible. Because the regular hexagon has angles of 120 degrees. If you try to divide it into equilateral triangles, which have 60-degree angles, you need to have specific divisions.

But the standard division is connecting the center to the vertices, which gives six equilateral triangles. Therefore, that must be the division they are referring to. Therefore, the answer is 42 inches.

Alternatively, maybe the problem is referring to the triangles formed by splitting the hexagon into smaller triangles. For example, if you divide each of the six equilateral triangles into smaller triangles, but then the number of triangles would be more than six. However, the problem says six triangles. Therefore, the only way is the standard division.

Therefore, I think my answer is correct. The perimeter of the hexagon is 42 inches.

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

To determine the perimeter of a regular hexagon that is divided into six equilateral triangles, we begin by analyzing the geometric relationship between the hexagon and the triangles.

---

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

A **regular hexagon** has six equal sides and six equal angles. When you connect the **center** of the hexagon to each of its **six vertices**, the hexagon is divided into **six congruent equilateral triangles**. Each triangle has:

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

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

---

### Step 2: Use the Given Perimeter of a Triangle

We are told that the **perimeter of one of these equilateral triangles is 21 inches**. Since all sides of an equilateral triangle are equal, we can find the length of one side by dividing the perimeter by 3:

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

This side length is also the **side length of the hexagon**, because each triangle shares one side with the hexagon.

---

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

Now that we know the side length of the hexagon is 7 inches, we can compute its perimeter by multiplying the side length by the number of sides:

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

---

### 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 the 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 for every three cookies, they get $1. So I need to figure out how many sets of three cookies they sold. Let me divide 54 by 3. 54 divided by 3 is 18. So that means they made 18 times $1 from the cookies. So 18 * $1 is $18. Okay, so cookies contribute $18 to their revenue.

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

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

Now, let me add up all the revenue from each item. So cookies: $18, cupcakes: $40, brownies: $35. Let me add those together. 18 + 40 is 58, and 58 + 35 is 93. So total revenue is $93?

Wait, let me check that again. 18 + 40 is 58? Yes, because 18 + 40 is 58. Then 58 + 35. Let me do 58 + 30 first, which is 88, then +5 is 93. Yeah, that seems right. So total revenue is $93.

Now, the cost to bake these items was $15. So profit is total revenue minus total cost. So that would be $93 minus $15. Let me compute that. 93 - 15. 93 - 10 is 83, then minus 5 more is 78. So profit is $78?

Wait, let me make sure I didn't make any mistakes in the calculations. Let me go through each step again.

Cookies: 54 cookies sold at three for $1. So 54 divided by 3 is indeed 18. 18 * $1 is $18. That seems right.

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

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

Total revenue: 18 + 40 + 35. Let me add them again. 18 + 40 is 58. 58 + 35. Let me think of 35 as 30 + 5. 58 + 30 is 88, plus 5 is 93. Yes, that's correct.

Total cost is $15. So profit is 93 - 15. 93 - 15. Let me do 93 - 10 = 83, then subtract 5 more: 83 - 5 = 78. So $78 profit. That seems straightforward.

Wait, but let me check if there's any other costs mentioned? The problem says "it cost the math club $15 to bake these items." So that's the only cost? So no other expenses? So profit is just total revenue minus that $15. So yes, 93 - 15 = 78. So the profit is $78.

Hmm. Let me think again if I interpreted the prices correctly. For the cookies, three for $1. So each cookie is $1/3? But they sold 54 cookies. So total revenue from cookies is (number of cookies) * (price per cookie). But since they sold them three for $1, the price per cookie is $1/3. So total revenue would be 54 * (1/3) = 18. Which is the same as before. So that's correct. So that's $18. So that's correct.

Alternatively, if you think of how many sets of three cookies they sold, which is 54 / 3 = 18 sets, each set sold for $1, so 18 * 1 = $18. Same result. So that's correct.

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

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

Total revenue: 18 + 40 + 35 = 93. Correct.

Total cost: 15. Profit: 93 - 15 = 78. So yes, $78. So I think that's the answer.

But let me check once more for any possible miscalculations. Maybe I added wrong? Let me add 18 + 40 + 35 again. 18 + 40 is 58. Then 58 + 35. Let me do 58 + 35. 58 + 30 is 88, 88 + 5 is 93. Correct. So total revenue is 93. Then 93 - 15. 93 - 10 is 83, 83 - 5 is 78. Correct. So profit is $78.

I think that's right. So the answer should be $\boxed{78}$.

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

To determine the math club's profit from the bake sale, we need to calculate the **total revenue** from the items sold and then subtract the **total cost** of baking those items.

---

### 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: Total Revenue

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

---

### Step 3: Subtract the Cost

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

---

### Step 4: Calculate Profit

Profit is total revenue minus total cost:
$$
93 - 15 = 78 \text{ dollars}
$$

---

### 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 the 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 for every three cookies, they get $1. So I need to figure out how many sets of three cookies they sold. Let me divide 54 by 3. 54 divided by 3 is 18. So that means they made 18 times $1 from the cookies. So 18 * $1 is $18. Okay, so cookies contribute $18 to their revenue.

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

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

Now, let me add up all the revenue from each item. So cookies: $18, cupcakes: $40, brownies: $35. Let me add those together. 18 + 40 is 58, and 58 + 35 is 93. So total revenue is $93?

Wait, let me check that again. 18 + 40 is 58? Yes, because 18 + 40 is 58. Then 58 + 35. Let me do 58 + 30 first, which is 88, then +5 is 93. Yeah, that seems right. So total revenue is $93.

Now, the cost to bake these items was $15. So profit is total revenue minus total cost. So that would be $93 minus $15. Let me compute that. 93 - 15. 93 - 10 is 83, then minus 5 more is 78. So profit is $78?

Wait, let me make sure I didn't make any mistakes in the calculations. Let me go through each step again.

Cookies: 54 cookies sold at three for $1. So 54 divided by 3 is indeed 18. 18 * $1 is $18. That seems right.

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

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

Total revenue: 18 + 40 + 35. Let me add them again. 18 + 40 is 58. 58 + 35. Let me think of 35 as 30 + 5. 58 + 30 is 88, plus 5 is 93. Yes, that's correct.

Total cost is $15. So profit is 93 - 15. 93 - 15. Let me do 93 - 10 = 83, then subtract 5 more: 83 - 5 = 78. So $78 profit. That seems straightforward.

Wait, but let me check if there's any other costs mentioned? The problem says "it cost the math club $15 to bake these items." So that's the only cost? So no other expenses? So profit is just total revenue minus that $15. So yes, 93 - 15 = 78. So the profit is $78.

Hmm. Let me think again if I interpreted the prices correctly. For the cookies, three for $1. So each cookie is $1/3? But they sold 54 cookies. So total revenue from cookies is (number of cookies) * (price per cookie). But since they sold them three for $1, the price per cookie is $1/3. So total revenue would be 54 * (1/3) = 18. Which is the same as before. So that's correct. So that's $18. So that's correct.

Alternatively, if you think of how many sets of three cookies they sold, which is 54 / 3 = 18 sets, each set sold for $1, so 18 * 1 = $18. Same result. So that's correct.

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

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

Total revenue: 18 + 40 + 35 = 93. Correct.

Total cost: 15. Profit: 93 - 15 = 78. So yes, $78. So I think that's the answer.

But let me check once more for any possible miscalculations. Maybe I added wrong? Let me add 18 + 40 + 35 again. 18 + 40 is 58. Then 58 + 35. Let me do 58 + 35. 58 + 30 is 88, 88 + 5 is 93. Correct. So total revenue is 93. Then 93 - 15. 93 - 10 is 83, 83 - 5 is 78. Correct. So profit is $78.

I think that's right. So the answer should be $\boxed{78}$.

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

To determine the math club's profit from the bake sale, we need to calculate the **total revenue** from the items sold and then subtract the **total cost** of baking those items.

---

### 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: Total Revenue

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

---

### Step 3: Subtract the Cost

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

---

### Step 4: Calculate Profit

Profit is total revenue minus total cost:
$$
93 - 15 = 78 \text{ dollars}
$$

---

### 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? Let me remember Euler's formula. 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 counter-clockwise around the origin would take it to (-b, a). Yes! So, that matches with the result of multiplying by i. So, the formula is correct.

So, 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.

First, 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. Let me check if that makes sense. The original point is (7, 2). Rotating 90 degrees counter-clockwise should move it to (-2, 7). Yes, that's exactly what -2 + 7i represents.

Wait, let me make sure I didn't mix up the coordinates. If you have a point (x, y), rotating 90 degrees counter-clockwise gives (-y, x). So original point is (7, 2), so after rotation, it's (-2, 7). Which is indeed -2 + 7i. So that seems correct.

Alternatively, maybe I can think of it in terms of matrices. A rotation matrix for 90 degrees counter-clockwise is [0, -1; 1, 0]. So, multiplying the vector [7; 2] by this matrix would give [0*7 + (-1)*2; 1*7 + 0*2] = [-2; 7], which is the same as -2 + 7i. Yep, same result.

So both methods give me the same answer. Therefore, I think that's correct.

But just to be thorough, let me think if there's another way to approach this. Maybe using polar form? Let me try that.

First, convert 7 + 2i to polar form. The modulus is sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53). The argument θ is arctangent(2/7). But when we rotate it by 90 degrees, we add π/2 to the argument. So the new argument is arctan(2/7) + π/2. Then, converting back to rectangular form... Hmm, that might be more complicated, but let me try.

Alternatively, since multiplying by i is equivalent to adding π/2 to the argument, so if we have modulus r and angle θ, after rotation it's modulus r and angle θ + π/2. So, converting back to rectangular coordinates, it's r*(cos(θ + π/2) + i sin(θ + π/2)).

But we can also use trigonometric identities. 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 r*(-sinθ + i cosθ).

But the original complex number is r*(cosθ + i sinθ). So, comparing, the rotated one is r*(-sinθ + i cosθ). Let me see if that's equivalent to multiplying by i.

Original: z = r(cosθ + i sinθ). Multiply by i: z*i = r(cosθ + i sinθ)*i = r(i cosθ - sinθ) = r(-sinθ + i cosθ). Which matches the above. So, yes, that's the same result.

Therefore, regardless of the method, multiplying by i gives the rotated complex number.

So, going back, original complex number is 7 + 2i. Multiply by i gives -2 + 7i.

Alternatively, if I didn't remember that multiplying by i rotates by 90 degrees, maybe I can think geometrically. The complex plane: the real axis is horizontal, imaginary is vertical. Rotating 90 degrees counter-clockwise would take the positive real axis to the positive imaginary axis, positive imaginary to negative real, etc.

So, the original point is 7 units right and 2 units up. Rotating 90 degrees counter-clockwise would make it 2 units left and 7 units up. Which is (-2, 7), so -2 + 7i. Yep, same answer.

Therefore, I think I can be confident that the answer is -2 + 7i.

But just to make sure, let me do a quick check with another example. Suppose I take a simple complex number, like 1 + 0i. Rotating it 90 degrees counter-clockwise should give 0 + 1i. Let's apply the formula: (1 + 0i)*i = i, which is 0 + 1i. Correct. Another example: 0 + 1i. Rotating 90 degrees counter-clockwise would take it to -1 + 0i. Multiply by i: (0 + 1i)*i = i² = -1, which is -1 + 0i. Correct.

Another example: 1 + i. Rotating 90 degrees counter-clockwise. The result should be -1 + i. Let's compute: (1 + i)*i = i + i² = i - 1 = -1 + i. Correct.

So, the method works for these examples. Therefore, applying it to 7 + 2i should be correct.

Therefore, the answer is -2 + 7i.

**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 a fundamental property of complex numbers: **multiplication by $i$** corresponds to a $90^\circ$ counter-clockwise rotation in the complex plane.

---

### Step 1: Understand the Rotation

In the complex plane, a complex number $z = a + bi$ can be represented as a point $(a, b)$. A $90^\circ$ counter-clockwise rotation around the origin transforms this point to $(-b, a)$. This transformation is equivalent to multiplying the complex number by $i$, since:

$$
i = 0 + 1i
$$

Multiplying a complex number $z = a + bi$ by $i$ gives:

$$
z \cdot i = (a + bi) \cdot i = ai + b i^2 = ai - b = -b + ai
$$

This matches the geometric transformation: $(a, b) \rightarrow (-b, a)$.

---

### Step 2: Apply the Rotation to $7 + 2i$

Let $z = 7 + 2i$. Multiply by $i$:

$$
(7 + 2i) \cdot i = 7i + 2i^2 = 7i + 2(-1) = 7i - 2 = -2 + 7i
$$

---

### 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 $(7, 2)$.

---

### 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? Let me remember Euler's formula. 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 counter-clockwise around the origin would take it to (-b, a). Yes! So, that matches with the result of multiplying by i. So, the formula is correct.

So, 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.

First, 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. Let me check if that makes sense. The original point is (7, 2). Rotating 90 degrees counter-clockwise should move it to (-2, 7). Yes, that's exactly what -2 + 7i represents.

Wait, let me make sure I didn't mix up the coordinates. If you have a point (x, y), rotating 90 degrees counter-clockwise gives (-y, x). So original point is (7, 2), so after rotation, it's (-2, 7). Which is indeed -2 + 7i. So that seems correct.

Alternatively, maybe I can think of it in terms of matrices. A rotation matrix for 90 degrees counter-clockwise is [0, -1; 1, 0]. So, multiplying the vector [7; 2] by this matrix would give [0*7 + (-1)*2; 1*7 + 0*2] = [-2; 7], which is the same as -2 + 7i. Yep, same result.

So both methods give me the same answer. Therefore, I think that's correct.

But just to be thorough, let me think if there's another way to approach this. Maybe using polar form? Let me try that.

First, convert 7 + 2i to polar form. The modulus is sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53). The argument θ is arctangent(2/7). But when we rotate it by 90 degrees, we add π/2 to the argument. So the new argument is arctan(2/7) + π/2. Then, converting back to rectangular form... Hmm, that might be more complicated, but let me try.

Alternatively, since multiplying by i is equivalent to adding π/2 to the argument, so if we have modulus r and angle θ, after rotation it's modulus r and angle θ + π/2. So, converting back to rectangular coordinates, it's r*(cos(θ + π/2) + i sin(θ + π/2)).

But we can also use trigonometric identities. 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 r*(-sinθ + i cosθ).

But the original complex number is r*(cosθ + i sinθ). So, comparing, the rotated one is r*(-sinθ + i cosθ). Let me see if that's equivalent to multiplying by i.

Original: z = r(cosθ + i sinθ). Multiply by i: z*i = r(cosθ + i sinθ)*i = r(i cosθ - sinθ) = r(-sinθ + i cosθ). Which matches the above. So, yes, that's the same result.

Therefore, regardless of the method, multiplying by i gives the rotated complex number.

So, going back, original complex number is 7 + 2i. Multiply by i gives -2 + 7i.

Alternatively, if I didn't remember that multiplying by i rotates by 90 degrees, maybe I can think geometrically. The complex plane: the real axis is horizontal, imaginary is vertical. Rotating 90 degrees counter-clockwise would take the positive real axis to the positive imaginary axis, positive imaginary to negative real, etc.

So, the original point is 7 units right and 2 units up. Rotating 90 degrees counter-clockwise would make it 2 units left and 7 units up. Which is (-2, 7), so -2 + 7i. Yep, same answer.

Therefore, I think I can be confident that the answer is -2 + 7i.

But just to make sure, let me do a quick check with another example. Suppose I take a simple complex number, like 1 + 0i. Rotating it 90 degrees counter-clockwise should give 0 + 1i. Let's apply the formula: (1 + 0i)*i = i, which is 0 + 1i. Correct. Another example: 0 + 1i. Rotating 90 degrees counter-clockwise would take it to -1 + 0i. Multiply by i: (0 + 1i)*i = i² = -1, which is -1 + 0i. Correct.

Another example: 1 + i. Rotating 90 degrees counter-clockwise. The result should be -1 + i. Let's compute: (1 + i)*i = i + i² = i - 1 = -1 + i. Correct.

So, the method works for these examples. Therefore, applying it to 7 + 2i should be correct.

Therefore, the answer is -2 + 7i.

**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 a fundamental property of complex numbers: **multiplication by $i$** corresponds to a $90^\circ$ counter-clockwise rotation in the complex plane.

---

### Step 1: Understand the Rotation

In the complex plane, a complex number $z = a + bi$ can be represented as a point $(a, b)$. A $90^\circ$ counter-clockwise rotation around the origin transforms this point to $(-b, a)$. This transformation is equivalent to multiplying the complex number by $i$, since:

$$
i = 0 + 1i
$$

Multiplying a complex number $z = a + bi$ by $i$ gives:

$$
z \cdot i = (a + bi) \cdot i = ai + b i^2 = ai - b = -b + ai
$$

This matches the geometric transformation: $(a, b) \rightarrow (-b, a)$.

---

### Step 2: Apply the Rotation to $7 + 2i$

Let $z = 7 + 2i$. Multiply by $i$:

$$
(7 + 2i) \cdot i = 7i + 2i^2 = 7i + 2(-1) = 7i - 2 = -2 + 7i
$$

---

### 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 $(7, 2)$.

---

### 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 $i(7 + 2i) = \boxed{-2 + 7i}.$