Explanation:
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, let me recall how that works. 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, let me remember the formulas for conversion. I think r is calculated using the Pythagorean theorem, right? So r equals the square root of x squared plus y squared. And 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 the given point is (0, 3). Let me visualize that. In the rectangular coordinate system, that would be on the positive y-axis, right? Because the x-coordinate is 0 and the y-coordinate is 3. So, this point is straight up along the y-axis.

Let me compute r first. Using the formula r = √(x² + y²). Plugging in x = 0 and y = 3, that would be √(0² + 3²) = √(0 + 9) = √9 = 3. So r is 3. That seems straightforward.

Now for θ. The formula is θ = arctan(y/x). But here, x is 0. Wait, arctan(3/0)... Hmm, division by zero. That's undefined. So how do I handle that? I think when x is 0, the angle is either π/2 or 3π/2, depending on the sign of y. Since our point is (0, 3), which is in the positive y-direction, that should be π/2 radians. Let me confirm.

In the unit circle, π/2 radians is 90 degrees, pointing straight up along the positive y-axis. Similarly, 3π/2 is 270 degrees, pointing straight down along the negative y-axis. Since our point is (0, 3), which is up the y-axis, θ should be π/2.

But let me think again. Is there another way to compute θ? Maybe using the arctangent with the two arguments? Wait, some calculators or programming languages have atan2(y, x) which takes into account the quadrant. In this case, since x is 0 and y is positive, atan2(3, 0) would give π/2. So that's correct.

But just to make sure, let's think about what θ represents. If you have a point on the y-axis, then the angle from the positive x-axis to the point is 90 degrees, which is π/2 radians. That's right.

So putting that all together, the polar coordinates should be (r, θ) = (3, π/2).

Wait, let me check if there's another possible angle. Since angles in polar coordinates are typically given between 0 and 2π, and r is supposed to be positive. If I tried to use a negative r, then θ would be different. But the problem specifies r > 0, so we can't use a negative r. Therefore, θ must be π/2.

Is there any other possibility? Let me think. If θ were π/2 + π = 3π/2, then that would point in the direction of the negative y-axis, but our point is on the positive y-axis. So 3π/2 would be incorrect. Therefore, π/2 is the correct angle.

Just to recap:

- r = √(0² + 3²) = 3

- θ = arctan(3/0) which is undefined, but since the point is on the positive y-axis, θ is π/2.

Yes, that seems right. Let me verify with another approach. If I imagine the point (0, 3), then in polar coordinates, moving 3 units from the origin along the direction of π/2 radians should bring me back to (0, 3).

Using the conversion from polar to rectangular coordinates, x = r cos θ and y = r sin θ. So if r is 3 and θ is π/2:

x = 3 cos(π/2) = 3 * 0 = 0

y = 3 sin(π/2) = 3 * 1 = 3

Which matches the original point (0, 3). Perfect, that checks out.

Therefore, the polar coordinates are (3, π/2).

I don't think there's any other angle in the range [0, 2π) that would satisfy this. Let me think if there's another angle with the same terminal side. But since the point is on the positive y-axis, the only angle in that range is π/2. Angles like 5π/2 would be equivalent, but 5π/2 is more than 2π, so it's not in the required range. Similarly, negative angles would be equivalent to positive ones by adding 2π, but they would not lie within 0 ≤ θ < 2π.

Therefore, I'm confident the answer is (3, π/2).

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

To convert the point $ (0, 3) $ from rectangular (Cartesian) coordinates to polar coordinates, we use the following relationships:

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

---

### Step 1: Calculate $ r $

Given:
$$
x = 0, \quad y = 3
$$

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

So, $ r = 3 $.

---

### Step 2: Determine $ \theta $

Next, compute the angle $ \theta $ using the arctangent function:
$$
\theta = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{3}{0}\right)
$$

This is undefined because division by zero occurs. However, we observe that the point $ (0, 3) $ lies directly on the **positive y-axis**.

In polar coordinates, angles are measured counterclockwise from the positive x-axis. The positive y-axis corresponds to an angle of $ \frac{\pi}{2} $ radians.

Thus:
$$
\theta = \frac{\pi}{2}
$$

---

### Step 3: Final Check

Let’s verify the conversion:
- $ x = r \cos \theta = 3 \cos\left(\frac{\pi}{2}\right) = 0 $
- $ y = r \sin \theta = 3 \sin\left(\frac{\pi}{2}\right) = 3 $

This confirms the original point $ (0, 3) $.

---

### Conclusion

The point $ (0, 3) $ in rectangular coordinates is equivalent to the polar coordinates:

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

[Assistant]
Explanation:
Okay, so I need to convert the rectangular coordinates (0, 3) to polar coordinates. Hmm, let me recall how that works. 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, let me remember the formulas for conversion. I think r is calculated using the Pythagorean theorem, right? So r equals the square root of x squared plus y squared. And 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 the given point is (0, 3). Let me visualize that. In the rectangular coordinate system, that would be on the positive y-axis, right? Because the x-coordinate is 0 and the y-coordinate is 3. So, this point is straight up along the y-axis.

Let me compute r first. Using the formula r = √(x² + y²). Plugging in x = 0 and y = 3, that would be √(0² + 3²) = √(0 + 9) = √9 = 3. So r is 3. That seems straightforward.

Now for θ. The formula is θ = arctan(y/x). But here, x is 0. Wait, arctan(3/0)... Hmm, division by zero. That's undefined. So how do I handle that? I think when x is 0, the angle is either π/2 or 3π/2, depending on the sign of y. Since our point is (0, 3), which is in the positive y-direction, that should be π/2 radians. Let me confirm.

In the unit circle, π/2 radians is 90 degrees, pointing straight up along the positive y-axis. Similarly, 3π/2 is 270 degrees, pointing straight down along the negative y-axis. Since our point is (0, 3), which is up the y-axis, θ should be π/2.

But let me think again. Is there another way to compute θ? Maybe using the arctangent with the two arguments? Wait, some calculators or programming languages have atan2(y, x) which takes into account the quadrant. In this case, since x is 0 and y is positive, atan2(3, 0) would give π/2. So that's correct.

But just to make sure, let's think about what θ represents. If you have a point on the y-axis, then the angle from the positive x-axis to the point is 90 degrees, which is π/2 radians. That's right.

So putting that all together, the polar coordinates should be (r, θ) = (3, π/2).

Wait, let me check if there's another possible angle. Since angles in polar coordinates are typically given between 0 and 2π, and r is supposed to be positive. If I tried to use a negative r, then θ would be different. But the problem specifies r > 0, so we can't use a negative r. Therefore, θ must be π/2.

Is there any other possibility? Let me think. If θ were π/2 + π = 3π/2, then that would point in the direction of the negative y-axis, but our point is on the positive y-axis. So 3π/2 would be incorrect. Therefore, π/2 is the correct angle.

Just to recap:

- r = √(0² + 3²) = 3

- θ = arctan(3/0) which is undefined, but since the point is on the positive y-axis, θ is π/2.

Yes, that seems right. Let me verify with another approach. If I imagine the point (0, 3), then in polar coordinates, moving 3 units from the origin along the direction of π/2 radians should bring me back to (0, 3).

Using the conversion from polar to rectangular coordinates, x = r cos θ and y = r sin θ. So if r is 3 and θ is π/2:

x = 3 cos(π/2) = 3 * 0 = 0

y = 3 sin(π/2) = 3 * 1 = 3

Which matches the original point (0, 3). Perfect, that checks out.

Therefore, the polar coordinates are (3, π/2).

I don't think there's any other angle in the range [0, 2π) that would satisfy this. Let me think if there's another angle with the same terminal side. But since the point is on the positive y-axis, the only angle in that range is π/2. Angles like 5π/2 would be equivalent, but 5π/2 is more than 2π, so it's not in the required range. Similarly, negative angles would be equivalent to positive ones by adding 2π, but they would not lie within 0 ≤ θ < 2π.

Therefore, I'm confident the answer is (3, π/2).

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

To convert the point $ (0, 3) $ from rectangular (Cartesian) coordinates to polar coordinates, we use the following relationships:

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

---

### Step 1: Calculate $ r $

Given:
$$
x = 0, \quad y = 3
$$

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

So, $ r = 3 $.

---

### Step 2: Determine $ \theta $

Next, compute the angle $ \theta $ using the arctangent function:
$$
\theta = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{3}{0}\right)
$$

This is undefined because division by zero occurs. However, we observe that the point $ (0, 3) $ lies directly on the **positive y-axis**.

In polar coordinates, angles are measured counterclockwise from the positive x-axis. The positive y-axis corresponds to an angle of $ \frac{\pi}{2} $ radians.

Thus:
$$
\theta = \frac{\pi}{2}
$$

---

### Step 3: Final Check

Let’s verify the conversion:
- $ x = r \cos \theta = 3 \cos\left(\frac{\pi}{2}\right) = 0 $
- $ y = r \sin \theta = 3 \sin\left(\frac{\pi}{2}\right) = 3 $

This confirms the original point $ (0, 3) $.

---

### Conclusion

The point $ (0, 3) $ in rectangular coordinates is equivalent to the polar coordinates:

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

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, basically, I need to find the distance each student traveled and the time they took, then compute their speeds and compare them.

Looking at the Asymptote code provided, there's a graph with time on the x-axis and distance on the y-axis. The graph has vertical lines at intervals from 0 to 7 on the x-axis (time) and horizontal lines from 0 to 6 on the y-axis (distance). The points are plotted as follows:

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

Each of these points represents the position of the student at a particular time. Wait, hold on—is this a position-time graph or a distance-time graph? The Asymptote code labels the y-axis as "distance" and the x-axis as "time," so it's a distance-time graph. In such graphs, the slope of the line represents the speed. However, since each student's position is plotted as a single point, I think the graph might be showing the position of each student at specific times? Or is it a cumulative distance?

Wait, maybe the graph is showing each student's position over time? But there are multiple students. Hmm. Wait, the Asymptote code is drawing the axes and then placing dots with labels for each student. So each dot is the position (distance) of the student at a certain time. Therefore, each student's data is a single point on the distance-time graph. So, to find the average speed for each student, we can calculate the distance from the starting point (origin) divided by the time it took them to reach that point.

But wait, hold on. If the graph is showing each student's position at a specific time, then for each student, their average speed would be the distance they've covered divided by the time they took to get there. However, since the graph is just a single point for each student, we need to assume that they started from the origin (O) at time 0 and reached their respective positions at their respective times. So the distance is the y-coordinate, and the time is the x-coordinate. Therefore, the average speed would be distance divided by time for each student.

Let me verify that understanding. If a student is at (1.25, 4.5), that would mean at time 1.25 units, they have covered 4.5 units of distance. Therefore, their average speed would be 4.5 divided by 1.25. Similarly for the others. So, let me compute each student's average speed.

First, let me list each student's time (x-coordinate) and distance (y-coordinate):

- Evelyn: time = 1.25, distance = 4.5
- Briana: time = 2.5, distance = 2.2
- Carla: time = 4.25, distance = 5.2
- Debra: time = 5.6, distance = 2.8
- Angela: time = 6.8, distance = 1.4

Now, compute average speed for each:

1. Evelyn: 4.5 / 1.25
Let me calculate that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 4.5 minus 3.75 is 0.75. 0.75 / 1.25 is 0.6. So total is 3.6. Alternatively, convert to fractions. 4.5 is 9/2, and 1.25 is 5/4. So (9/2) divided by (5/4) is (9/2)*(4/5) = 36/10 = 3.6. So Evelyn's speed is 3.6 units per time unit.

2. Briana: 2.2 / 2.5

Let me compute that. 2.2 divided by 2.5. Multiply numerator and denominator by 10: 22 / 25. That is 0.88. So Briana's speed is 0.88 units per time unit.

3. Carla: 5.2 / 4.25

Let me calculate this. 5.2 divided by 4.25. Let me convert to fractions or decimals. 4.25 is 17/4, and 5.2 is 52/10 = 26/5. So (26/5) divided by (17/4) is (26/5)*(4/17) = (104)/85 ≈ 1.2235 units per time unit. Alternatively, decimal: 4.25 goes into 5.2 how many times? 4.25 * 1 = 4.25, subtract that from 5.2, you get 0.95. Then, 0.95 / 4.25 = 0.2235... So total is approximately 1.2235. So around 1.22 units per time.

4. Debra: 2.8 / 5.6. That's straightforward. 2.8 divided by 5.6 is 0.5. So Debra's speed is 0.5 units per time.

5. Angela: 1.4 / 6.8. Let me compute that. 1.4 divided by 6.8. Multiply numerator and denominator by 10: 14 / 68 = 7 / 34 ≈ 0.2059 units per time.

So summarizing:

- Evelyn: 3.6
- Briana: 0.88
- Carla: ≈1.22
- Debra: 0.5
- Angela: ≈0.2059

Comparing these values, Evelyn has the highest average speed at 3.6, followed by Carla with ~1.22, then Briana with 0.88, Debra with 0.5, and Angela with ~0.2059.

Therefore, the student with the greatest average speed is Evelyn.

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

Starting with Evelyn: 4.5 divided by 1.25. Let me do decimal division. 1.25 goes into 4.5. Let me think of it as 1.25 * 3 = 3.75, subtract that from 4.5, gives 0.75. Then, 1.25 goes into 0.75 0.6 times (because 1.25 * 0.6 = 0.75). So total is 3 + 0.6 = 3.6. Correct.

Briana: 2.2 / 2.5. Let me think of it as 2.2 divided by 2.5. Well, dividing by 2.5 is the same as multiplying by 0.4. So 2.2 * 0.4 = 0.88. Correct.

Carla: 5.2 / 4.25. Let me convert 4.25 to 4.25. If I multiply numerator and denominator by 4, to eliminate decimals. 5.2 * 4 = 20.8, 4.25 * 4 = 17. So 20.8 / 17. Let me divide 20.8 by 17. 17 goes into 20 once, remainder 3.8. 17 goes into 3.8 0.2235... times. So total is 1.2235. Correct.

Debra: 2.8 / 5.6. Well, 2.8 is half of 5.6, so 2.8 / 5.6 = 0.5. Correct.

Angela: 1.4 / 6.8. Let me compute 1.4 divided by 6.8. 1.4 divided by 6.8. Let me multiply numerator and denominator by 10: 14 / 68. Simplify by dividing numerator and denominator by 2: 7 / 34 ≈ 0.20588. Correct.

So all calculations check out. Evelyn has the highest average speed of 3.6, so Evelyn is the answer.

But just to make sure I didn't misinterpret the graph. The key assumption here is that each student started at the origin (0,0) and their position is given at a specific time. Therefore, the distance they traveled is the y-coordinate, and the time is the x-coordinate. Therefore, average speed is distance over time. This seems correct because average speed is total distance over total time. Since they started at 0, their total distance is the y-coordinate, and total time is x-coordinate. Therefore, the calculation is correct.

Alternatively, if the graph was showing something else, like displacement vs. time, but the question says it's a distance-time graph, so distance is the dependent variable. So yes, the calculation is correct.

Therefore, the answer should be Evelyn.

**Final Answer**
The student with the greatest average speed is \boxed{Evelyn}.

To determine which student has the greatest average speed based on the given graph, we begin by understanding the nature of the graph: it is a **distance-time graph**, where the x-axis represents **time** and the y-axis represents **distance**. Each student's position is represented by a point on the graph, which gives us:

- Time at which the student was at a given position
- The corresponding distance covered by that time

Average speed is defined as:

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

Using the coordinates of each student, we calculate their average speed:

---

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

---

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

---

**Carla**: (4.25, 5.2)
$$
\text{Average Speed} = \frac{5.2}{4.25} \approx 1.22
$$

---

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

---

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

---

### Comparing the Speeds

Let's list the average speeds again:

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

From this comparison, **Evelyn** clearly has the **greatest 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, basically, I need to find the distance each student traveled and the time they took, then compute their speeds and compare them.

Looking at the Asymptote code provided, there's a graph with time on the x-axis and distance on the y-axis. The graph has vertical lines at intervals from 0 to 7 on the x-axis (time) and horizontal lines from 0 to 6 on the y-axis (distance). The points are plotted as follows:

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

Each of these points represents the position of the student at a particular time. Wait, hold on—is this a position-time graph or a distance-time graph? The Asymptote code labels the y-axis as "distance" and the x-axis as "time," so it's a distance-time graph. In such graphs, the slope of the line represents the speed. However, since each student's position is plotted as a single point, I think the graph might be showing the position of each student at specific times? Or is it a cumulative distance?

Wait, maybe the graph is showing each student's position over time? But there are multiple students. Hmm. Wait, the Asymptote code is drawing the axes and then placing dots with labels for each student. So each dot is the position (distance) of the student at a certain time. Therefore, each student's data is a single point on the distance-time graph. So, to find the average speed for each student, we can calculate the distance from the starting point (origin) divided by the time it took them to reach that point.

But wait, hold on. If the graph is showing each student's position at a specific time, then for each student, their average speed would be the distance they've covered divided by the time they took to get there. However, since the graph is just a single point for each student, we need to assume that they started from the origin (O) at time 0 and reached their respective positions at their respective times. So the distance is the y-coordinate, and the time is the x-coordinate. Therefore, the average speed would be distance divided by time for each student.

Let me verify that understanding. If a student is at (1.25, 4.5), that would mean at time 1.25 units, they have covered 4.5 units of distance. Therefore, their average speed would be 4.5 divided by 1.25. Similarly for the others. So, let me compute each student's average speed.

First, let me list each student's time (x-coordinate) and distance (y-coordinate):

- Evelyn: time = 1.25, distance = 4.5
- Briana: time = 2.5, distance = 2.2
- Carla: time = 4.25, distance = 5.2
- Debra: time = 5.6, distance = 2.8
- Angela: time = 6.8, distance = 1.4

Now, compute average speed for each:

1. Evelyn: 4.5 / 1.25
Let me calculate that. 4.5 divided by 1.25. Well, 1.25 times 3 is 3.75, and 4.5 minus 3.75 is 0.75. 0.75 / 1.25 is 0.6. So total is 3.6. Alternatively, convert to fractions. 4.5 is 9/2, and 1.25 is 5/4. So (9/2) divided by (5/4) is (9/2)*(4/5) = 36/10 = 3.6. So Evelyn's speed is 3.6 units per time unit.

2. Briana: 2.2 / 2.5

Let me compute that. 2.2 divided by 2.5. Multiply numerator and denominator by 10: 22 / 25. That is 0.88. So Briana's speed is 0.88 units per time unit.

3. Carla: 5.2 / 4.25

Let me calculate this. 5.2 divided by 4.25. Let me convert to fractions or decimals. 4.25 is 17/4, and 5.2 is 52/10 = 26/5. So (26/5) divided by (17/4) is (26/5)*(4/17) = (104)/85 ≈ 1.2235 units per time unit. Alternatively, decimal: 4.25 goes into 5.2 how many times? 4.25 * 1 = 4.25, subtract that from 5.2, you get 0.95. Then, 0.95 / 4.25 = 0.2235... So total is approximately 1.2235. So around 1.22 units per time.

4. Debra: 2.8 / 5.6. That's straightforward. 2.8 divided by 5.6 is 0.5. So Debra's speed is 0.5 units per time.

5. Angela: 1.4 / 6.8. Let me compute that. 1.4 divided by 6.8. Multiply numerator and denominator by 10: 14 / 68 = 7 / 34 ≈ 0.2059 units per time.

So summarizing:

- Evelyn: 3.6
- Briana: 0.88
- Carla: ≈1.22
- Debra: 0.5
- Angela: ≈0.2059

Comparing these values, Evelyn has the highest average speed at 3.6, followed by Carla with ~1.22, then Briana with 0.88, Debra with 0.5, and Angela with ~0.2059.

Therefore, the student with the greatest average speed is Evelyn.

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

Starting with Evelyn: 4.5 divided by 1.25. Let me do decimal division. 1.25 goes into 4.5. Let me think of it as 1.25 * 3 = 3.75, subtract that from 4.5, gives 0.75. Then, 1.25 goes into 0.75 0.6 times (because 1.25 * 0.6 = 0.75). So total is 3 + 0.6 = 3.6. Correct.

Briana: 2.2 / 2.5. Let me think of it as 2.2 divided by 2.5. Well, dividing by 2.5 is the same as multiplying by 0.4. So 2.2 * 0.4 = 0.88. Correct.

Carla: 5.2 / 4.25. Let me convert 4.25 to 4.25. If I multiply numerator and denominator by 4, to eliminate decimals. 5.2 * 4 = 20.8, 4.25 * 4 = 17. So 20.8 / 17. Let me divide 20.8 by 17. 17 goes into 20 once, remainder 3.8. 17 goes into 3.8 0.2235... times. So total is 1.2235. Correct.

Debra: 2.8 / 5.6. Well, 2.8 is half of 5.6, so 2.8 / 5.6 = 0.5. Correct.

Angela: 1.4 / 6.8. Let me compute 1.4 divided by 6.8. 1.4 divided by 6.8. Let me multiply numerator and denominator by 10: 14 / 68. Simplify by dividing numerator and denominator by 2: 7 / 34 ≈ 0.20588. Correct.

So all calculations check out. Evelyn has the highest average speed of 3.6, so Evelyn is the answer.

But just to make sure I didn't misinterpret the graph. The key assumption here is that each student started at the origin (0,0) and their position is given at a specific time. Therefore, the distance they traveled is the y-coordinate, and the time is the x-coordinate. Therefore, average speed is distance over time. This seems correct because average speed is total distance over total time. Since they started at 0, their total distance is the y-coordinate, and total time is x-coordinate. Therefore, the calculation is correct.

Alternatively, if the graph was showing something else, like displacement vs. time, but the question says it's a distance-time graph, so distance is the dependent variable. So yes, the calculation is correct.

Therefore, the answer should be Evelyn.

**Final Answer**
The student with the greatest average speed is \boxed{Evelyn}.

To determine which student has the greatest average speed based on the given graph, we begin by understanding the nature of the graph: it is a **distance-time graph**, where the x-axis represents **time** and the y-axis represents **distance**. Each student's position is represented by a point on the graph, which gives us:

- Time at which the student was at a given position
- The corresponding distance covered by that time

Average speed is defined as:

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

Using the coordinates of each student, we calculate their average speed:

---

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

---

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

---

**Carla**: (4.25, 5.2)
$$
\text{Average Speed} = \frac{5.2}{4.25} \approx 1.22
$$

---

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

---

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

---

### Comparing the Speeds

Let's list the average speeds again:

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

From this comparison, **Evelyn** clearly has the **greatest 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 been divided into six equilateral triangles. They told me that the perimeter of one of those triangles is 21 inches. Let me try to visualize this first. A regular hexagon has six equal sides and angles. If you draw lines from the center to each of the six vertices, it divides the hexagon into six congruent equilateral triangles. So each of those triangles has sides that are equal in length.

Wait, so if the perimeter of one of these triangles is 21 inches, then each side of the triangle must be... Well, an equilateral triangle has three sides of equal length. So if the perimeter is 21, then each side would be 21 divided by 3. Let me write that down:

Perimeter of triangle = 3 × side length
So, side length = Perimeter / 3
Which would be 21 / 3 = 7 inches.

So each side of the equilateral triangle is 7 inches. Now, how does that relate to the hexagon? Since the hexagon is divided into six such triangles, each side of the hexagon should be equal to the side length of the triangle. Wait, let me think again. Let me draw a mental picture. In a regular hexagon, each side is equal, and when you divide it into six equilateral triangles by connecting the center to each vertex, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. Wait, so if the triangles are equilateral, then all sides are equal. Therefore, the length of the side of the hexagon is equal to the radius of the hexagon. So the sides of the triangle are equal to the sides of the hexagon.

Therefore, each side of the hexagon is 7 inches. Since a regular hexagon has six sides, the perimeter of the hexagon is 6 × side length. So that would be 6 × 7. Let me calculate that: 6 × 7 = 42 inches.

Wait, but let me verify that. If each triangle has sides of 7 inches, then the side of the hexagon is 7 inches. Therefore, the perimeter is 6 × 7 = 42. That seems straightforward. But let me make sure that I didn't confuse the sides. Let me think again.

Imagine one of those triangles. The base of the triangle is a side of the hexagon. The other two sides of the triangle are from the center of the hexagon to each vertex. Since the triangle is equilateral, all three sides must be equal. Therefore, the distance from the center to a vertex (which is the radius of the circumscribed circle) is equal to the length of the side of the hexagon. Therefore, the side length of the hexagon is equal to the side length of the triangle. Therefore, yes, the perimeter of the hexagon would be 6 times 7, which is 42.

But let me check if there's another interpretation. Suppose the perimeter of the triangle is 21 inches. If the triangle is formed by dividing the hexagon, maybe the sides of the triangle are not the same as the sides of the hexagon? Wait, but if the hexagon is regular, then all sides are equal. When you split it into six triangles, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. Since it's an equilateral triangle, then those radii must be equal in length to the sides of the hexagon. Therefore, the two sides of the triangle that are radii are equal to the side of the hexagon. Therefore, each side of the triangle is equal to the side of the hexagon. Therefore, perimeter of triangle is 3 × side of hexagon. Therefore, side of hexagon is 7 inches. Then perimeter is 6 × 7 = 42.

Alternatively, maybe someone might think that the triangle's perimeter includes the radius, but since the triangle is equilateral, all sides are equal. So if the triangle is composed of two radii and one side of the hexagon, then all those are equal, meaning the side of the hexagon is equal to the radius.

Wait, but maybe there's confusion between the radius and the side length? Let me recall properties of regular hexagons. In a regular hexagon, the side length is equal to the radius of the circumscribed circle. That is, the distance from the center to any vertex is equal to the length of each side. Therefore, if we have a triangle formed by two radii and a side of the hexagon, since all sides are equal, it is an equilateral triangle. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if the perimeter of the triangle is 21 inches, then 3 × side length = 21 inches, so side length = 7 inches. Then perimeter of the hexagon is 6 × side length = 6 × 7 = 42 inches.

Therefore, I think that's correct. Wait, but let me think if there's another way that the triangle could be formed. For example, maybe the triangle isn't formed by connecting the center to the vertices, but perhaps another division? But the problem says a regular hexagon can be divided into six equilateral triangles. So the standard way is connecting the center to each vertex, forming six equilateral triangles.

Alternatively, could the triangles be formed differently? For instance, maybe dividing the hexagon into smaller triangles? But if you divide a regular hexagon into six equilateral triangles, the most straightforward way is connecting the center.

Alternatively, if you divide it into six congruent equilateral triangles, then yes, each triangle would have two sides that are radii and the base as a side of the hexagon.

Therefore, with that, I think my reasoning is correct.

But just to make sure, let me think about the length again. Suppose each side of the triangle is 7 inches. Then the side of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6×7=42.

Alternatively, if I thought that the triangle's perimeter was referring to something else, but I don't think so. The problem says "the perimeter of one of the triangles is 21 inches." Since each triangle is equilateral, then their sides are all equal. Therefore, each side is 7 inches. Therefore, the hexagon's perimeter is 6×7.

Therefore, the answer should be 42 inches.

But let me just check with an example. Let's say the side length of the hexagon is 's'. Then each of those triangles has sides of length 's', so their perimeter is 3s. If 3s = 21, then s=7. Therefore, perimeter of hexagon is 6s=42. Yep.

Alternatively, if someone thought that the triangles were different, for example, if they thought that the triangles were maybe with sides of the hexagon and other lengths, but no, since the triangles are equilateral, they have to have all sides equal. Therefore, the triangles must have sides equal to the side of the hexagon.

Therefore, the perimeter of the hexagon is 42 inches.

I think that makes sense. So I think the answer is 42.

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

To determine the perimeter of a regular hexagon, we are given that the hexagon can be divided into **six equilateral triangles**, and that the **perimeter of one of these triangles is 21 inches**.

---

### Step 1: Understand the Triangle's Side Length

Each triangle is **equilateral**, meaning all three sides are equal in length. If the **perimeter** of one triangle is 21 inches, then the length of one side of the triangle is:

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

---

### Step 2: Relate the Triangle to the Hexagon

A regular hexagon is made of six congruent **equilateral triangles** formed by connecting the **center of the hexagon to its six vertices**. As a result:

- Each triangle has **two sides** that are **radii** of the hexagon (i.e., distance from center to a vertex).
- The **third side** is one **side of the hexagon**.

Since the triangle is equilateral, all three sides (including the side of the hexagon) are equal in length. Therefore:

$$
\text{Side length of hexagon} = 7 \text{ inches}
$$

---

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

A regular hexagon has **six equal sides**. So, the 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 been divided into six equilateral triangles. They told me that the perimeter of one of those triangles is 21 inches. Let me try to visualize this first. A regular hexagon has six equal sides and angles. If you draw lines from the center to each of the six vertices, it divides the hexagon into six congruent equilateral triangles. So each of those triangles has sides that are equal in length.

Wait, so if the perimeter of one of these triangles is 21 inches, then each side of the triangle must be... Well, an equilateral triangle has three sides of equal length. So if the perimeter is 21, then each side would be 21 divided by 3. Let me write that down:

Perimeter of triangle = 3 × side length
So, side length = Perimeter / 3
Which would be 21 / 3 = 7 inches.

So each side of the equilateral triangle is 7 inches. Now, how does that relate to the hexagon? Since the hexagon is divided into six such triangles, each side of the hexagon should be equal to the side length of the triangle. Wait, let me think again. Let me draw a mental picture. In a regular hexagon, each side is equal, and when you divide it into six equilateral triangles by connecting the center to each vertex, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. Wait, so if the triangles are equilateral, then all sides are equal. Therefore, the length of the side of the hexagon is equal to the radius of the hexagon. So the sides of the triangle are equal to the sides of the hexagon.

Therefore, each side of the hexagon is 7 inches. Since a regular hexagon has six sides, the perimeter of the hexagon is 6 × side length. So that would be 6 × 7. Let me calculate that: 6 × 7 = 42 inches.

Wait, but let me verify that. If each triangle has sides of 7 inches, then the side of the hexagon is 7 inches. Therefore, the perimeter is 6 × 7 = 42. That seems straightforward. But let me make sure that I didn't confuse the sides. Let me think again.

Imagine one of those triangles. The base of the triangle is a side of the hexagon. The other two sides of the triangle are from the center of the hexagon to each vertex. Since the triangle is equilateral, all three sides must be equal. Therefore, the distance from the center to a vertex (which is the radius of the circumscribed circle) is equal to the length of the side of the hexagon. Therefore, the side length of the hexagon is equal to the side length of the triangle. Therefore, yes, the perimeter of the hexagon would be 6 times 7, which is 42.

But let me check if there's another interpretation. Suppose the perimeter of the triangle is 21 inches. If the triangle is formed by dividing the hexagon, maybe the sides of the triangle are not the same as the sides of the hexagon? Wait, but if the hexagon is regular, then all sides are equal. When you split it into six triangles, each triangle has two sides that are radii of the hexagon and one side that is a side of the hexagon. Since it's an equilateral triangle, then those radii must be equal in length to the sides of the hexagon. Therefore, the two sides of the triangle that are radii are equal to the side of the hexagon. Therefore, each side of the triangle is equal to the side of the hexagon. Therefore, perimeter of triangle is 3 × side of hexagon. Therefore, side of hexagon is 7 inches. Then perimeter is 6 × 7 = 42.

Alternatively, maybe someone might think that the triangle's perimeter includes the radius, but since the triangle is equilateral, all sides are equal. So if the triangle is composed of two radii and one side of the hexagon, then all those are equal, meaning the side of the hexagon is equal to the radius.

Wait, but maybe there's confusion between the radius and the side length? Let me recall properties of regular hexagons. In a regular hexagon, the side length is equal to the radius of the circumscribed circle. That is, the distance from the center to any vertex is equal to the length of each side. Therefore, if we have a triangle formed by two radii and a side of the hexagon, since all sides are equal, it is an equilateral triangle. Therefore, the perimeter of each triangle is 3 times the side length of the hexagon. Therefore, if the perimeter of the triangle is 21 inches, then 3 × side length = 21 inches, so side length = 7 inches. Then perimeter of the hexagon is 6 × side length = 6 × 7 = 42 inches.

Therefore, I think that's correct. Wait, but let me think if there's another way that the triangle could be formed. For example, maybe the triangle isn't formed by connecting the center to the vertices, but perhaps another division? But the problem says a regular hexagon can be divided into six equilateral triangles. So the standard way is connecting the center to each vertex, forming six equilateral triangles.

Alternatively, could the triangles be formed differently? For instance, maybe dividing the hexagon into smaller triangles? But if you divide a regular hexagon into six equilateral triangles, the most straightforward way is connecting the center.

Alternatively, if you divide it into six congruent equilateral triangles, then yes, each triangle would have two sides that are radii and the base as a side of the hexagon.

Therefore, with that, I think my reasoning is correct.

But just to make sure, let me think about the length again. Suppose each side of the triangle is 7 inches. Then the side of the hexagon is 7 inches. Therefore, the perimeter of the hexagon is 6×7=42.

Alternatively, if I thought that the triangle's perimeter was referring to something else, but I don't think so. The problem says "the perimeter of one of the triangles is 21 inches." Since each triangle is equilateral, then their sides are all equal. Therefore, each side is 7 inches. Therefore, the hexagon's perimeter is 6×7.

Therefore, the answer should be 42 inches.

But let me just check with an example. Let's say the side length of the hexagon is 's'. Then each of those triangles has sides of length 's', so their perimeter is 3s. If 3s = 21, then s=7. Therefore, perimeter of hexagon is 6s=42. Yep.

Alternatively, if someone thought that the triangles were different, for example, if they thought that the triangles were maybe with sides of the hexagon and other lengths, but no, since the triangles are equilateral, they have to have all sides equal. Therefore, the triangles must have sides equal to the side of the hexagon.

Therefore, the perimeter of the hexagon is 42 inches.

I think that makes sense. So I think the answer is 42.

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

To determine the perimeter of a regular hexagon, we are given that the hexagon can be divided into **six equilateral triangles**, and that the **perimeter of one of these triangles is 21 inches**.

---

### Step 1: Understand the Triangle's Side Length

Each triangle is **equilateral**, meaning all three sides are equal in length. If the **perimeter** of one triangle is 21 inches, then the length of one side of the triangle is:

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

---

### Step 2: Relate the Triangle to the Hexagon

A regular hexagon is made of six congruent **equilateral triangles** formed by connecting the **center of the hexagon to its six vertices**. As a result:

- Each triangle has **two sides** that are **radii** of the hexagon (i.e., distance from center to a vertex).
- The **third side** is one **side of the hexagon**.

Since the triangle is equilateral, all three sides (including the side of the hexagon) are equal in length. Therefore:

$$
\text{Side length of hexagon} = 7 \text{ inches}
$$

---

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

A regular hexagon has **six equal sides**. So, the 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:

V = πr²h

Alright, so the volume V is 45π, and they gave me the radius, right? Wait, looking back at the problem, there's an Asymptote figure. Let me check that again. The Asymptote code draws a cylinder with some labels. It mentions that the radius is 3 cm. Let me confirm. The Asymptote code labels "r=3" between the points (2.2,0) and (3.4,0), so that segment is the radius. So the radius is 3 centimeters. Got it.

So, plugging the values into the volume formula:

45π = π * r² * h

Since the radius r is 3 cm, substitute that in:

45π = π * (3)² * h

Calculating 3 squared is 9, so:

45π = π * 9 * h

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

45 = 9 * h

Okay, so then to solve for h, divide both sides by 9:

h = 45 / 9

Which simplifies to:

h = 5

Wait, so the height is 5 centimeters? Let me just check my steps again to make sure I didn't make a mistake.

First, volume formula: correct. Then substituting the given volume and radius: correct. Calculated 3 squared: correct. Divided both sides by π: correct. Then divided 45 by 9: yes, that's 5. Seems straightforward.

But let me check if there's any trick here. The Asymptote code might have some scaling or something? Wait, the Asymptote code is just a diagram, right? The problem says "the cylinder shown," so the diagram is just illustrative. The important thing is that they told us the radius is 3 cm. The Asymptote code might have some scaling factors, but since they labeled the radius as 3, we don't need to worry about the diagram's coordinates.

So, yeah, I think my calculation is correct. Volume equals πr²h, plug in the numbers, solve for h, get 5. Let me check with another approach. If radius is 3, then area of the base is πr² = π*9. Then volume is base area times height, so 9π*h = 45π. Divide both sides by 9π, get h = 45π / 9π = 5. Yep, same answer.

Therefore, the height is 5 centimeters. I think that's it. No mistakes here.

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

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

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

We are given:
- Volume $ V = 45\pi $ cubic centimeters
- Radius $ r = 3 $ centimeters

---

### Step 1: Plug the known values into the formula

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

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

---

### Step 2: Simplify the equation

We can 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:

V = πr²h

Alright, so the volume V is 45π, and they gave me the radius, right? Wait, looking back at the problem, there's an Asymptote figure. Let me check that again. The Asymptote code draws a cylinder with some labels. It mentions that the radius is 3 cm. Let me confirm. The Asymptote code labels "r=3" between the points (2.2,0) and (3.4,0), so that segment is the radius. So the radius is 3 centimeters. Got it.

So, plugging the values into the volume formula:

45π = π * r² * h

Since the radius r is 3 cm, substitute that in:

45π = π * (3)² * h

Calculating 3 squared is 9, so:

45π = π * 9 * h

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

45 = 9 * h

Okay, so then to solve for h, divide both sides by 9:

h = 45 / 9

Which simplifies to:

h = 5

Wait, so the height is 5 centimeters? Let me just check my steps again to make sure I didn't make a mistake.

First, volume formula: correct. Then substituting the given volume and radius: correct. Calculated 3 squared: correct. Divided both sides by π: correct. Then divided 45 by 9: yes, that's 5. Seems straightforward.

But let me check if there's any trick here. The Asymptote code might have some scaling or something? Wait, the Asymptote code is just a diagram, right? The problem says "the cylinder shown," so the diagram is just illustrative. The important thing is that they told us the radius is 3 cm. The Asymptote code might have some scaling factors, but since they labeled the radius as 3, we don't need to worry about the diagram's coordinates.

So, yeah, I think my calculation is correct. Volume equals πr²h, plug in the numbers, solve for h, get 5. Let me check with another approach. If radius is 3, then area of the base is πr² = π*9. Then volume is base area times height, so 9π*h = 45π. Divide both sides by 9π, get h = 45π / 9π = 5. Yep, same answer.

Therefore, the height is 5 centimeters. I think that's it. No mistakes here.

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

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

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

We are given:
- Volume $ V = 45\pi $ cubic centimeters
- Radius $ r = 3 $ centimeters

---

### Step 1: Plug the known values into the formula

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

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

---

### Step 2: Simplify the equation

We can 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, so the math club is having a bake sale to raise money for a trip. They sold some cookies, cupcakes, and brownies, and we need to figure out their profit after subtracting the cost of baking. Let me break this down step by step.

First, let me recall what profit means. Profit is calculated by subtracting the total costs from the total revenue. So, I need to find out how much money they made from selling the items (revenue) and then subtract the cost of baking ($15) to get the profit. Got it.

Alright, let me start with the cookies. They sold 54 cookies at three for $1. Hmm, so I need to figure out how much money they made from the cookies. If three cookies cost $1, then the price per cookie is $1 divided by 3. Let me compute that: $1 ÷ 3 ≈ $0.333... per cookie. But maybe it's easier to calculate how many sets of three cookies they sold.

So total cookies sold is 54. If each set is three cookies, then the number of sets is 54 ÷ 3. Let me do that division: 54 divided by 3 is 18. So they sold 18 sets of three cookies. Since each set is $1, then the total revenue from cookies is 18 × $1 = $18. Okay, that seems straightforward. So cookies contribute $18 to the revenue.

Next up are the cupcakes. They sold 20 cupcakes at $2 each. So that's straightforward multiplication. 20 cupcakes × $2 each. Let me calculate that: 20 × 2 = $40. So cupcakes bring in $40.

Then there are the brownies. They sold 35 brownies at $1 each. That's simple too: 35 × $1 = $35. So brownies contribute $35.

Now, let me add up all the revenue from each item. Cookies: $18, cupcakes: $40, brownies: $35. Adding them together: 18 + 40 is 58, and then 58 + 35. Let me do that step by step. 58 + 30 is 88, and then +5 is 93. So total revenue is $93?

Wait, hold on. Let me check that again. 18 (cookies) + 40 (cupcakes) is indeed 58. Then 58 + 35: 58 + 35. 58 + 30 is 88, plus 5 is 93. Yep, so total revenue is $93.

But let me verify the cookie calculation once more because that seems like a place where mistakes can happen. They sold 54 cookies at three for a dollar. So 54 divided by 3 is 18, so 18 dollars. That seems right. So 18 + 40 + 35. Yeah, 18 + 40 is 58, 58 + 35 is 93. Okay, so revenue is $93.

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

Wait, that seems straightforward, but let me check again to be sure.

Alternatively, I can write down each part again:

Cookies: 54 cookies, 3 for $1. Number of groups of three: 54 / 3 = 18. So 18 * $1 = $18. Correct.

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

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

Total revenue: 18 + 40 + 35. Let me add 18 + 40 first. 18 + 40 = 58. Then 58 + 35. 58 + 35. Breaking it down, 58 + 35 = 58 + 30 + 5 = 88 + 5 = 93. So $93 total revenue.

Total cost is $15. Therefore, profit is 93 - 15. Let me compute that again. 93 - 15. 93 - 10 = 83. 83 - 5 = 78. So yes, $78 profit.

But let me think again—did I miss anything? For example, are there any other costs? The problem says "it cost the math club $15 to bake these items," so that's the only cost mentioned. So we don't have to consider anything else? Like, maybe they had other expenses? But the problem says "it cost the math club $15 to bake these items," so probably that's the only cost. So profit is indeed revenue minus $15. So $93 - $15 = $78.

Alternatively, maybe I should check if there's a different interpretation of the problem. For example, when they say "three for $1," is that per set or per cookie? But the problem says "three for $1," so that would mean that each set of three cookies is sold for $1. So yes, that's how we calculated it. So 54 cookies divided into sets of three is 18 sets, so 18 dollars.

Alternatively, if someone thought that three for $1 means each cookie is $1/3, then 54 cookies would be 54*(1/3) = $18. So same result. So either way, it's $18. So that's correct.

Cupcakes are straightforward, 20 at $2, so $40. Brownies 35 at $1, so $35. Total revenue $93, subtract $15 cost, profit $78.

Wait, but let me check once more if I added correctly. 18 + 40 is 58, 58 + 35. Let me add 18 + 35 first. 18 + 35 is 53, then +40 is 93. Yeah, same result.

Alternatively, maybe the problem is in the units? Like, is the $15 cost included in the revenue? No, because the problem says "it cost the math club $15 to bake these items," so that's an expense, so subtract that from the revenue.

So yeah, I think that's all. Unless there's a trick or something I missed, but the problem seems straightforward.

So, profit is $78. Therefore, the answer is $\boxed{78}$.

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

To determine the profit made by the math club from their bake sale, we first calculate the **total revenue** from the items sold and then subtract the **total cost** of baking the items.

---

### Step 1: Calculate Revenue from Each Item

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

#### **Cupcakes**
- **Quantity sold:** 20 cupcakes
- **Price per cupcake:** \$2
- **Revenue from cupcakes:**
  $$
  20 \times 2 = \$40
  $$

#### **Brownies**
- **Quantity sold:** 35 brownies
- **Price per brownie:** \$1
- **Revenue from brownies:**
  $$
  35 \times 1 = \$35
  $$

---

### Step 2: Total Revenue

Now we add up the revenues from each item:
$$
18 + 40 + 35 = 93
$$
So, total **revenue** is \$93.

---

### Step 3: Subtract the Cost to Compute Profit

The total cost to bake the items is given as:
$$
\$15
$$

Thus, the **profit** is:
$$
\text{Profit} = \text{Revenue} - \text{Cost} = 93 - 15 = \boxed{78}
$$

---

### ✅ Final Answer

The math club's profit was $\boxed{78}$ dollars.

[Assistant]
Explanation:
Okay, so the math club is having a bake sale to raise money for a trip. They sold some cookies, cupcakes, and brownies, and we need to figure out their profit after subtracting the cost of baking. Let me break this down step by step.

First, let me recall what profit means. Profit is calculated by subtracting the total costs from the total revenue. So, I need to find out how much money they made from selling the items (revenue) and then subtract the cost of baking ($15) to get the profit. Got it.

Alright, let me start with the cookies. They sold 54 cookies at three for $1. Hmm, so I need to figure out how much money they made from the cookies. If three cookies cost $1, then the price per cookie is $1 divided by 3. Let me compute that: $1 ÷ 3 ≈ $0.333... per cookie. But maybe it's easier to calculate how many sets of three cookies they sold.

So total cookies sold is 54. If each set is three cookies, then the number of sets is 54 ÷ 3. Let me do that division: 54 divided by 3 is 18. So they sold 18 sets of three cookies. Since each set is $1, then the total revenue from cookies is 18 × $1 = $18. Okay, that seems straightforward. So cookies contribute $18 to the revenue.

Next up are the cupcakes. They sold 20 cupcakes at $2 each. So that's straightforward multiplication. 20 cupcakes × $2 each. Let me calculate that: 20 × 2 = $40. So cupcakes bring in $40.

Then there are the brownies. They sold 35 brownies at $1 each. That's simple too: 35 × $1 = $35. So brownies contribute $35.

Now, let me add up all the revenue from each item. Cookies: $18, cupcakes: $40, brownies: $35. Adding them together: 18 + 40 is 58, and then 58 + 35. Let me do that step by step. 58 + 30 is 88, and then +5 is 93. So total revenue is $93?

Wait, hold on. Let me check that again. 18 (cookies) + 40 (cupcakes) is indeed 58. Then 58 + 35: 58 + 35. 58 + 30 is 88, plus 5 is 93. Yep, so total revenue is $93.

But let me verify the cookie calculation once more because that seems like a place where mistakes can happen. They sold 54 cookies at three for a dollar. So 54 divided by 3 is 18, so 18 dollars. That seems right. So 18 + 40 + 35. Yeah, 18 + 40 is 58, 58 + 35 is 93. Okay, so revenue is $93.

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

Wait, that seems straightforward, but let me check again to be sure.

Alternatively, I can write down each part again:

Cookies: 54 cookies, 3 for $1. Number of groups of three: 54 / 3 = 18. So 18 * $1 = $18. Correct.

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

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

Total revenue: 18 + 40 + 35. Let me add 18 + 40 first. 18 + 40 = 58. Then 58 + 35. 58 + 35. Breaking it down, 58 + 35 = 58 + 30 + 5 = 88 + 5 = 93. So $93 total revenue.

Total cost is $15. Therefore, profit is 93 - 15. Let me compute that again. 93 - 15. 93 - 10 = 83. 83 - 5 = 78. So yes, $78 profit.

But let me think again—did I miss anything? For example, are there any other costs? The problem says "it cost the math club $15 to bake these items," so that's the only cost mentioned. So we don't have to consider anything else? Like, maybe they had other expenses? But the problem says "it cost the math club $15 to bake these items," so probably that's the only cost. So profit is indeed revenue minus $15. So $93 - $15 = $78.

Alternatively, maybe I should check if there's a different interpretation of the problem. For example, when they say "three for $1," is that per set or per cookie? But the problem says "three for $1," so that would mean that each set of three cookies is sold for $1. So yes, that's how we calculated it. So 54 cookies divided into sets of three is 18 sets, so 18 dollars.

Alternatively, if someone thought that three for $1 means each cookie is $1/3, then 54 cookies would be 54*(1/3) = $18. So same result. So either way, it's $18. So that's correct.

Cupcakes are straightforward, 20 at $2, so $40. Brownies 35 at $1, so $35. Total revenue $93, subtract $15 cost, profit $78.

Wait, but let me check once more if I added correctly. 18 + 40 is 58, 58 + 35. Let me add 18 + 35 first. 18 + 35 is 53, then +40 is 93. Yeah, same result.

Alternatively, maybe the problem is in the units? Like, is the $15 cost included in the revenue? No, because the problem says "it cost the math club $15 to bake these items," so that's an expense, so subtract that from the revenue.

So yeah, I think that's all. Unless there's a trick or something I missed, but the problem seems straightforward.

So, profit is $78. Therefore, the answer is $\boxed{78}$.

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

To determine the profit made by the math club from their bake sale, we first calculate the **total revenue** from the items sold and then subtract the **total cost** of baking the items.

---

### Step 1: Calculate Revenue from Each Item

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

#### **Cupcakes**
- **Quantity sold:** 20 cupcakes
- **Price per cupcake:** \$2
- **Revenue from cupcakes:**
  $$
  20 \times 2 = \$40
  $$

#### **Brownies**
- **Quantity sold:** 35 brownies
- **Price per brownie:** \$1
- **Revenue from brownies:**
  $$
  35 \times 1 = \$35
  $$

---

### Step 2: Total Revenue

Now we add up the revenues from each item:
$$
18 + 40 + 35 = 93
$$
So, total **revenue** is \$93.

---

### Step 3: Subtract the Cost to Compute Profit

The total cost to bake the items is given as:
$$
\$15
$$

Thus, the **profit** is:
$$
\text{Profit} = \text{Revenue} - \text{Cost} = 93 - 15 = \boxed{78}
$$

---

### ✅ Final Answer

The math club's profit was $\boxed{78}$ dollars.

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 represented using multiplication by another complex number. Let me recall...

I think rotating a complex number by an angle θ is done by multiplying it by e^(iθ). Since we're dealing with a 90-degree rotation, θ is 90 degrees, which is π/2 radians. So, e^(iπ/2) is cos(π/2) + i sin(π/2). Let me calculate that. Cos(π/2) is 0, and sin(π/2) is 1, so e^(iπ/2) is 0 + i*1 = i.

Wait, so multiplying the original complex number by i should give me the rotated one? Let me check that with a simple example. If I take a complex number like 1 + 0i, multiplying by i would give i, which is a 90-degree rotation from the positive real axis to the positive imaginary axis. That works. What about another number? Let's say 1 + i. Rotating that 90 degrees counter-clockwise... The original point is at (1,1). After a 90-degree rotation, it should be at (-1, 1), right? Let's see: (1 + i) * i = i + i² = i - 1 = -1 + i. Which is (-1, 1). Yep, that matches. So the multiplication by i does indeed rotate the complex number 90 degrees counter-clockwise.

So, applying that to 7 + 2i, I just need to multiply it by i. Let me write that out:

(7 + 2i) * i.

Let me compute that step by step. Distribute the multiplication:

7 * i + 2i * i = 7i + 2i².

Now, since i² is -1, this becomes:

7i + 2*(-1) = 7i - 2.

Which can be written as -2 + 7i.

Therefore, the resulting complex number is -2 + 7i.

Wait, let me verify this another way. Maybe using the rotation formula. Another way to think about rotating a complex number is using coordinates. If a complex number is represented as (x, y), then rotating it 90 degrees counter-clockwise around the origin gives (-y, x). Let me check that.

Original point: 7 + 2i corresponds to (7, 2). Rotating 90 degrees counter-clockwise would give (-2, 7), which is -2 + 7i. Exactly the same result as before! So that confirms it.

Therefore, both methods give me the same answer. So I can be confident that multiplying by i or using the coordinate transformation gives me -2 + 7i.

Is there another way to think about this? Maybe using matrices? Rotations can be represented by rotation matrices. The rotation matrix for 90 degrees counter-clockwise is:

[ 0, -1 ]
[ 1,  0 ]

So, multiplying this matrix by the vector [7, 2] should give the rotated vector. Let me do that:

First component: 0*7 + (-1)*2 = -2

Second component: 1*7 + 0*2 = 7

So the resulting vector is [-2, 7], which is again -2 + 7i.

So all methods lead to the same result. Therefore, I think that's the correct answer.

Just to make sure I didn't make any calculation mistakes. Let me go over the multiplication again:

Original complex number: 7 + 2i

Multiply by i:

First, 7 * i = 7i

Then, 2i * i = 2i² = 2*(-1) = -2

Adding them together: 7i - 2, which is indeed -2 + 7i.

Yes, that seems right.

Alternatively, maybe I can visualize it. The original complex number 7 + 2i is in the first quadrant, 7 units along the real axis and 2 units up. Rotating it 90 degrees counter-clockwise should move it to the second quadrant, where the real part becomes negative and the imaginary part becomes positive. Which is exactly what -2 + 7i is. So, real part -2, imaginary part 7. That makes sense.

Another check: The modulus of the original complex number should be the same after rotation, since rotation is a rigid transformation. Let me verify that.

Original modulus: sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53)

After rotation: modulus is sqrt((-2)^2 + 7^2) = sqrt(4 + 49) = sqrt(53). Same modulus. So that checks out.

Also, the argument (angle) of the original complex number is arctangent(2/7). After rotation, it should be arctangent(2/7) + 90 degrees. Let me check the argument of the rotated number.

The rotated number is -2 + 7i. Its argument is arctangent(7 / (-2)), but since it's in the second quadrant, we add 180 degrees. But arctangent(7 / (-2)) is arctangent(-3.5), which is in the fourth quadrant. However, adjusting for the quadrant, the actual angle is 180 - arctangent(7/2). Which is exactly the original angle plus 90 degrees? Wait, let's see.

Original angle θ = arctan(2/7). After adding 90 degrees, the new angle is θ + 90 degrees. Let's check if arctan(7/2) is θ + 90 degrees? Wait, maybe I need to think more carefully.

Let me recall that if you have a complex number z = x + iy, then its argument is arctan(y/x), but adjusted for the quadrant. After rotating by 90 degrees, the new complex number is -y + ix. Its argument is arctan( (x)/(-y) ) adjusted for quadrant. Wait, let me take specific numbers.

Original z = 7 + 2i, so x = 7, y = 2. After rotation, it becomes -2 + 7i, so the new x is -2, new y is 7. The argument is arctan(7 / (-2)), but since it's in the second quadrant, the angle is π - arctan(7/2). Let me check if this is equal to original angle + π/2.

Original angle θ = arctan(2/7). Let's compute θ + π/2.

But tan(θ + π/2) is undefined, because tan(θ + π/2) = -cotθ. Wait, maybe it's easier to check numerically. Let me compute θ ≈ arctan(2/7). Let me calculate that. 2/7 ≈ 0.2857. arctan(0.2857) ≈ 16.1 degrees. Then θ + 90 degrees ≈ 106.1 degrees.

Now, the new angle is π - arctan(7/2). Let me compute arctan(7/2). 7/2 is 3.5. arctan(3.5) ≈ 74 degrees. Therefore, π - 74 degrees ≈ 180 - 74 = 106 degrees. Which matches θ + 90 degrees (approx 16.1 + 90 = 106.1 degrees). So yes, that checks out. Therefore, the angle is indeed increased by 90 degrees, which is what we expect from a rotation.

So all these checks confirm that the result is correct.

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

But just to recap all steps once more:

1. Recognize that rotation by 90 degrees counter-clockwise is equivalent to multiplying by i.

2. Multiply 7 + 2i by i: (7 + 2i) * i = 7i + 2i² = 7i - 2.

3. Rearranged as -2 + 7i.

4. Verified through coordinate transformation, rotation matrix, modulus check, and angle check.

All methods agree, so the answer is correct.

I think that's thorough enough. I don't see any mistakes in my reasoning.

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

To solve the problem of rotating the complex number $ 7 + 2i $ by $ 90^\circ $ **counter-clockwise** about the origin, we rely on a fundamental property of complex numbers and rotations in the complex plane.

---

### Step 1: Understanding the Rotation in the Complex Plane

A **rotation** of a complex number in the plane by an angle $ \theta $ counterclockwise can be represented by **multiplying** the complex number by the complex number $ e^{i\theta} $.

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

---

### Step 2: Apply the Rotation

We are rotating the complex number $ 7 + 2i $, so we multiply it by $ i $:
$$
(7 + 2i) \cdot i
$$

Distribute the multiplication:
$$
7 \cdot i + 2i \cdot i = 7i + 2i^2
$$

Recall that $ i^2 = -1 $, so:
$$
7i + 2(-1) = 7i - 2
$$

Rewriting:
$$
-2 + 7i
$$

---

### Step 3: Verification via Coordinate Representation

We can also think of the complex number as a point in the plane: $ (7, 2) $. A $ 90^\circ $ counterclockwise rotation about the origin maps a point $ (x, y) $ to $ (-y, x) $. So:
$$
(7, 2) \mapsto (-2, 7)
$$

Which again corresponds to the complex number $ -2 + 7i $, confirming the result.

---

### Final Result

All approaches — algebraic, geometric, and through the properties of rotation — agree that the result of a $ 90^\circ $ counter-clockwise rotation of $ 7 + 2i $ is:

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

[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 represented using multiplication by another complex number. Let me recall...

I think rotating a complex number by an angle θ is done by multiplying it by e^(iθ). Since we're dealing with a 90-degree rotation, θ is 90 degrees, which is π/2 radians. So, e^(iπ/2) is cos(π/2) + i sin(π/2). Let me calculate that. Cos(π/2) is 0, and sin(π/2) is 1, so e^(iπ/2) is 0 + i*1 = i.

Wait, so multiplying the original complex number by i should give me the rotated one? Let me check that with a simple example. If I take a complex number like 1 + 0i, multiplying by i would give i, which is a 90-degree rotation from the positive real axis to the positive imaginary axis. That works. What about another number? Let's say 1 + i. Rotating that 90 degrees counter-clockwise... The original point is at (1,1). After a 90-degree rotation, it should be at (-1, 1), right? Let's see: (1 + i) * i = i + i² = i - 1 = -1 + i. Which is (-1, 1). Yep, that matches. So the multiplication by i does indeed rotate the complex number 90 degrees counter-clockwise.

So, applying that to 7 + 2i, I just need to multiply it by i. Let me write that out:

(7 + 2i) * i.

Let me compute that step by step. Distribute the multiplication:

7 * i + 2i * i = 7i + 2i².

Now, since i² is -1, this becomes:

7i + 2*(-1) = 7i - 2.

Which can be written as -2 + 7i.

Therefore, the resulting complex number is -2 + 7i.

Wait, let me verify this another way. Maybe using the rotation formula. Another way to think about rotating a complex number is using coordinates. If a complex number is represented as (x, y), then rotating it 90 degrees counter-clockwise around the origin gives (-y, x). Let me check that.

Original point: 7 + 2i corresponds to (7, 2). Rotating 90 degrees counter-clockwise would give (-2, 7), which is -2 + 7i. Exactly the same result as before! So that confirms it.

Therefore, both methods give me the same answer. So I can be confident that multiplying by i or using the coordinate transformation gives me -2 + 7i.

Is there another way to think about this? Maybe using matrices? Rotations can be represented by rotation matrices. The rotation matrix for 90 degrees counter-clockwise is:

[ 0, -1 ]
[ 1,  0 ]

So, multiplying this matrix by the vector [7, 2] should give the rotated vector. Let me do that:

First component: 0*7 + (-1)*2 = -2

Second component: 1*7 + 0*2 = 7

So the resulting vector is [-2, 7], which is again -2 + 7i.

So all methods lead to the same result. Therefore, I think that's the correct answer.

Just to make sure I didn't make any calculation mistakes. Let me go over the multiplication again:

Original complex number: 7 + 2i

Multiply by i:

First, 7 * i = 7i

Then, 2i * i = 2i² = 2*(-1) = -2

Adding them together: 7i - 2, which is indeed -2 + 7i.

Yes, that seems right.

Alternatively, maybe I can visualize it. The original complex number 7 + 2i is in the first quadrant, 7 units along the real axis and 2 units up. Rotating it 90 degrees counter-clockwise should move it to the second quadrant, where the real part becomes negative and the imaginary part becomes positive. Which is exactly what -2 + 7i is. So, real part -2, imaginary part 7. That makes sense.

Another check: The modulus of the original complex number should be the same after rotation, since rotation is a rigid transformation. Let me verify that.

Original modulus: sqrt(7² + 2²) = sqrt(49 + 4) = sqrt(53)

After rotation: modulus is sqrt((-2)^2 + 7^2) = sqrt(4 + 49) = sqrt(53). Same modulus. So that checks out.

Also, the argument (angle) of the original complex number is arctangent(2/7). After rotation, it should be arctangent(2/7) + 90 degrees. Let me check the argument of the rotated number.

The rotated number is -2 + 7i. Its argument is arctangent(7 / (-2)), but since it's in the second quadrant, we add 180 degrees. But arctangent(7 / (-2)) is arctangent(-3.5), which is in the fourth quadrant. However, adjusting for the quadrant, the actual angle is 180 - arctangent(7/2). Which is exactly the original angle plus 90 degrees? Wait, let's see.

Original angle θ = arctan(2/7). After adding 90 degrees, the new angle is θ + 90 degrees. Let's check if arctan(7/2) is θ + 90 degrees? Wait, maybe I need to think more carefully.

Let me recall that if you have a complex number z = x + iy, then its argument is arctan(y/x), but adjusted for the quadrant. After rotating by 90 degrees, the new complex number is -y + ix. Its argument is arctan( (x)/(-y) ) adjusted for quadrant. Wait, let me take specific numbers.

Original z = 7 + 2i, so x = 7, y = 2. After rotation, it becomes -2 + 7i, so the new x is -2, new y is 7. The argument is arctan(7 / (-2)), but since it's in the second quadrant, the angle is π - arctan(7/2). Let me check if this is equal to original angle + π/2.

Original angle θ = arctan(2/7). Let's compute θ + π/2.

But tan(θ + π/2) is undefined, because tan(θ + π/2) = -cotθ. Wait, maybe it's easier to check numerically. Let me compute θ ≈ arctan(2/7). Let me calculate that. 2/7 ≈ 0.2857. arctan(0.2857) ≈ 16.1 degrees. Then θ + 90 degrees ≈ 106.1 degrees.

Now, the new angle is π - arctan(7/2). Let me compute arctan(7/2). 7/2 is 3.5. arctan(3.5) ≈ 74 degrees. Therefore, π - 74 degrees ≈ 180 - 74 = 106 degrees. Which matches θ + 90 degrees (approx 16.1 + 90 = 106.1 degrees). So yes, that checks out. Therefore, the angle is indeed increased by 90 degrees, which is what we expect from a rotation.

So all these checks confirm that the result is correct.

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

But just to recap all steps once more:

1. Recognize that rotation by 90 degrees counter-clockwise is equivalent to multiplying by i.

2. Multiply 7 + 2i by i: (7 + 2i) * i = 7i + 2i² = 7i - 2.

3. Rearranged as -2 + 7i.

4. Verified through coordinate transformation, rotation matrix, modulus check, and angle check.

All methods agree, so the answer is correct.

I think that's thorough enough. I don't see any mistakes in my reasoning.

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

To solve the problem of rotating the complex number $ 7 + 2i $ by $ 90^\circ $ **counter-clockwise** about the origin, we rely on a fundamental property of complex numbers and rotations in the complex plane.

---

### Step 1: Understanding the Rotation in the Complex Plane

A **rotation** of a complex number in the plane by an angle $ \theta $ counterclockwise can be represented by **multiplying** the complex number by the complex number $ e^{i\theta} $.

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

---

### Step 2: Apply the Rotation

We are rotating the complex number $ 7 + 2i $, so we multiply it by $ i $:
$$
(7 + 2i) \cdot i
$$

Distribute the multiplication:
$$
7 \cdot i + 2i \cdot i = 7i + 2i^2
$$

Recall that $ i^2 = -1 $, so:
$$
7i + 2(-1) = 7i - 2
$$

Rewriting:
$$
-2 + 7i
$$

---

### Step 3: Verification via Coordinate Representation

We can also think of the complex number as a point in the plane: $ (7, 2) $. A $ 90^\circ $ counterclockwise rotation about the origin maps a point $ (x, y) $ to $ (-y, x) $. So:
$$
(7, 2) \mapsto (-2, 7)
$$

Which again corresponds to the complex number $ -2 + 7i $, confirming the result.

---

### Final Result

All approaches — algebraic, geometric, and through the properties of rotation — agree that the result of a $ 90^\circ $ counter-clockwise rotation of $ 7 + 2i $ is:

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

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


Explanation:
Okay, let's see. The problem is about figuring out how many different battalions can be sent to the Battle of Thermopylae based on the number of upper and lower class soldiers available. Let me break it down step by step.

First, let me make sure I understand the problem correctly. There are two types of soldiers in the Greek army: upper class and lower class. In a certain part of Athens, there are 5 upper class soldiers and 10 lower class soldiers. The battle requires a force of 4 upper class soldiers and 8 lower class soldiers. The question is asking how many different battalions can be sent.

Hmm, so I think this is a combinations problem where we need to select a certain number of soldiers from each class to form a battalion that meets the required force. But wait, the problem says "different battalions," so I need to figure out how many unique combinations of upper and lower class soldiers can be formed while meeting the required numbers for the battle. Wait, actually, the battle demands a specific force: 4 upper and 8 lower. So does that mean that each battalion that is sent must consist exactly of 4 upper and 8 lower class soldiers? Or is the battle demanding that the total force has at least 4 upper and 8 lower? Wait, the wording says "the battle of Thermopylae demands a force of 4 upper class soldiers and 8 lower class soldiers." So I think that means that each battalion sent must have exactly 4 upper and 8 lower class soldiers. Therefore, the question is asking how many different battalions can be formed with exactly 4 upper and 8 lower class soldiers, given that there are 5 upper and 10 lower class soldiers available.

Wait, but if that's the case, then it's just the number of ways to choose 4 upper class soldiers out of 5 and 8 lower class soldiers out of 10. Then the total number of different battalions would be the product of those two combinations. Let me check that reasoning.

So, for the upper class soldiers: there are 5 available, and we need to choose 4. The number of ways to do that is "5 choose 4", which is calculated as 5! / (4! * (5-4)!) = 5.

For the lower class soldiers: there are 10 available, and we need to choose 8. The number of ways to do that is "10 choose 8", which is 10! / (8! * (10-8)!) = 10! / (8! * 2!) = (10 * 9 * 8! ) / (8! * 2 * 1) ) = (10 * 9)/2 = 45.

Therefore, the total number of different battalions would be 5 (ways to choose upper class) multiplied by 45 (ways to choose lower class), which is 5 * 45 = 225.

Wait, so is the answer 225? But let me think again. Is there a possibility that the problem is not asking for exactly 4 upper and 8 lower, but rather any number of upper and lower soldiers as long as they meet the demand? But the wording says "the battle demands a force of 4 upper class soldiers and 8 lower class soldiers". So I think that means that each battalion sent must consist of exactly those numbers. Otherwise, if they could have more, the problem would probably say something like "at least 4 upper and 8 lower". So I think my original reasoning holds.

But just to be thorough, let's consider alternative interpretations. Suppose the problem was asking how many different battalions can be formed where each battalion is a force that meets the battle's requirement, which is 4 upper and 8 lower. Therefore, each battalion must be exactly that number. So the number of different battalions is the number of ways to choose 4 upper from 5 and 8 lower from 10, which is 5 * 45 = 225.

Alternatively, if the problem was asking for how many different combinations of upper and lower class soldiers can be sent, where the total force meets the requirement (i.e., 4 upper and 8 lower), then it's the same as above. So yes, 225.

But let me make sure there isn't a trick here. The problem says "different battalions". Maybe the question is considering that different combinations of soldiers are different battalions, even if they have the same number of upper and lower soldiers? But no, the question specifies that the battle requires 4 upper and 8 lower, so each battalion must have exactly those numbers. Therefore, the different battalions are different based on which soldiers are selected.

So, for example, if you have 5 upper class soldiers, say labeled A, B, C, D, E, then choosing A, B, C, D is different from choosing A, B, C, E, etc. Similarly for the lower class soldiers. Therefore, the total number is indeed combinations of upper and lower soldiers.

Therefore, the answer is 5 choose 4 times 10 choose 8, which is 5 * 45 = 225.

Wait, but let me check the calculation again for 10 choose 8. 10 choose 8 is equal to 10 choose 2 because n choose k is equal to n choose (n - k). So 10 choose 2 is 45, which matches. So that's correct.

Similarly, 5 choose 4 is 5, which is correct.

Therefore, multiplying them gives 225.

But just to make sure, maybe the problem is in some other way? For example, maybe the battalions are considered different if they have different numbers of soldiers? But the problem says "the battle demands a force of 4 upper class soldiers and 8 lower class soldiers", so the battalion must be exactly that size. So you can't have more or less. Therefore, the answer is 225.

Alternatively, if the question had asked how many different battalions can be formed with any number of soldiers as long as they meet the battle's requirement (i.e., at least 4 upper and 8 lower), then the answer would be different. For example, you could have 4 upper and 8 lower, 5 upper and 8 lower, etc. But the problem specifically says "a force of 4 upper class soldiers and 8 lower class soldiers", which implies exactly that number.

So, I think the answer is 225.

Just to check once more: total upper soldiers available: 5. Need to choose 4. Number of ways: 5. Lower soldiers: 10, need to choose 8: 45. Multiply them: 5 * 45 = 225.

Alternatively, maybe the problem is a trick question where they consider that the battalion is determined by the number of upper and lower soldiers, not the specific individuals? But that seems unlikely. The term "different battalions" would usually imply different groupings of soldiers.

For example, if you have two different groups of soldiers, even if they have the same number of upper and lower, they are different battalions if the members are different.

Therefore, the answer should be 225.

Therefore, I think that is the correct answer.

**Final Answer**
The number of different battalions that can be sent is \boxed{225}.

To determine how many different battalions can be sent to the Battle of Thermopylae, we need to analyze the selection of soldiers from two classes — upper and lower — based on the required composition of the battalion.

---

### **Step 1: Understand the Composition Requirement**
The battle demands a force composed of:
- **4 upper class soldiers**
- **8 lower class soldiers**

This means that **each battalion sent must be composed of exactly 4 upper class and 8 lower class soldiers**. The problem does not allow for any flexibility in this composition.

---

### **Step 2: Determine the Available Soldiers**
There are:
- **5 upper class soldiers**
- **10 lower class soldiers**

Thus, we are selecting 4 out of 5 upper class soldiers, and 8 out of 10 lower class soldiers.

---

### **Step 3: Use Combinations to Count Possible selections**

#### **Upper Class Soldiers**
We calculate how many ways we can choose 4 soldiers out of 5. This is a standard combination problem:

$$
\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5
$$

#### **Lower Class Soldiers**
Next, we calculate how many ways we can choose 8 out of 10 soldiers:

$$
\binom{10}{8} = \binom{10}{2} = \frac{10 \times 9}{2} = 45
$$

---

### **Step 4: Multiply the Two Combinations**
Since the choice of upper class and lower class soldiers is independent, the total number of distinct battalions that can be formed is the product of the two combinations:

$$
\text{Total Battalions} = \binom{5}{4} \times \binom{10}{8} = 5 \times 45 = 225
$$

---

### **Final Answer**
$$
\boxed{225}
$$

[Assistant]
Explanation:
Okay, let's see. The problem is about figuring out how many different battalions can be sent to the Battle of Thermopylae based on the number of upper and lower class soldiers available. Let me break it down step by step.

First, let me make sure I understand the problem correctly. There are two types of soldiers in the Greek army: upper class and lower class. In a certain part of Athens, there are 5 upper class soldiers and 10 lower class soldiers. The battle requires a force of 4 upper class soldiers and 8 lower class soldiers. The question is asking how many different battalions can be sent.

Hmm, so I think this is a combinations problem where we need to select a certain number of soldiers from each class to form a battalion that meets the required force. But wait, the problem says "different battalions," so I need to figure out how many unique combinations of upper and lower class soldiers can be formed while meeting the required numbers for the battle. Wait, actually, the battle demands a specific force: 4 upper and 8 lower. So does that mean that each battalion that is sent must consist exactly of 4 upper and 8 lower class soldiers? Or is the battle demanding that the total force has at least 4 upper and 8 lower? Wait, the wording says "the battle of Thermopylae demands a force of 4 upper class soldiers and 8 lower class soldiers." So I think that means that each battalion sent must have exactly 4 upper and 8 lower class soldiers. Therefore, the question is asking how many different battalions can be formed with exactly 4 upper and 8 lower class soldiers, given that there are 5 upper and 10 lower class soldiers available.

Wait, but if that's the case, then it's just the number of ways to choose 4 upper class soldiers out of 5 and 8 lower class soldiers out of 10. Then the total number of different battalions would be the product of those two combinations. Let me check that reasoning.

So, for the upper class soldiers: there are 5 available, and we need to choose 4. The number of ways to do that is "5 choose 4", which is calculated as 5! / (4! * (5-4)!) = 5.

For the lower class soldiers: there are 10 available, and we need to choose 8. The number of ways to do that is "10 choose 8", which is 10! / (8! * (10-8)!) = 10! / (8! * 2!) = (10 * 9 * 8! ) / (8! * 2 * 1) ) = (10 * 9)/2 = 45.

Therefore, the total number of different battalions would be 5 (ways to choose upper class) multiplied by 45 (ways to choose lower class), which is 5 * 45 = 225.

Wait, so is the answer 225? But let me think again. Is there a possibility that the problem is not asking for exactly 4 upper and 8 lower, but rather any number of upper and lower soldiers as long as they meet the demand? But the wording says "the battle demands a force of 4 upper class soldiers and 8 lower class soldiers". So I think that means that each battalion sent must consist of exactly those numbers. Otherwise, if they could have more, the problem would probably say something like "at least 4 upper and 8 lower". So I think my original reasoning holds.

But just to be thorough, let's consider alternative interpretations. Suppose the problem was asking how many different battalions can be formed where each battalion is a force that meets the battle's requirement, which is 4 upper and 8 lower. Therefore, each battalion must be exactly that number. So the number of different battalions is the number of ways to choose 4 upper from 5 and 8 lower from 10, which is 5 * 45 = 225.

Alternatively, if the problem was asking for how many different combinations of upper and lower class soldiers can be sent, where the total force meets the requirement (i.e., 4 upper and 8 lower), then it's the same as above. So yes, 225.

But let me make sure there isn't a trick here. The problem says "different battalions". Maybe the question is considering that different combinations of soldiers are different battalions, even if they have the same number of upper and lower soldiers? But no, the question specifies that the battle requires 4 upper and 8 lower, so each battalion must have exactly those numbers. Therefore, the different battalions are different based on which soldiers are selected.

So, for example, if you have 5 upper class soldiers, say labeled A, B, C, D, E, then choosing A, B, C, D is different from choosing A, B, C, E, etc. Similarly for the lower class soldiers. Therefore, the total number is indeed combinations of upper and lower soldiers.

Therefore, the answer is 5 choose 4 times 10 choose 8, which is 5 * 45 = 225.

Wait, but let me check the calculation again for 10 choose 8. 10 choose 8 is equal to 10 choose 2 because n choose k is equal to n choose (n - k). So 10 choose 2 is 45, which matches. So that's correct.

Similarly, 5 choose 4 is 5, which is correct.

Therefore, multiplying them gives 225.

But just to make sure, maybe the problem is in some other way? For example, maybe the battalions are considered different if they have different numbers of soldiers? But the problem says "the battle demands a force of 4 upper class soldiers and 8 lower class soldiers", so the battalion must be exactly that size. So you can't have more or less. Therefore, the answer is 225.

Alternatively, if the question had asked how many different battalions can be formed with any number of soldiers as long as they meet the battle's requirement (i.e., at least 4 upper and 8 lower), then the answer would be different. For example, you could have 4 upper and 8 lower, 5 upper and 8 lower, etc. But the problem specifically says "a force of 4 upper class soldiers and 8 lower class soldiers", which implies exactly that number.

So, I think the answer is 225.

Just to check once more: total upper soldiers available: 5. Need to choose 4. Number of ways: 5. Lower soldiers: 10, need to choose 8: 45. Multiply them: 5 * 45 = 225.

Alternatively, maybe the problem is a trick question where they consider that the battalion is determined by the number of upper and lower soldiers, not the specific individuals? But that seems unlikely. The term "different battalions" would usually imply different groupings of soldiers.

For example, if you have two different groups of soldiers, even if they have the same number of upper and lower, they are different battalions if the members are different.

Therefore, the answer should be 225.

Therefore, I think that is the correct answer.

**Final Answer**
The number of different battalions that can be sent is \boxed{225}.

To determine how many different battalions can be sent to the Battle of Thermopylae, we need to analyze the selection of soldiers from two classes — upper and lower — based on the required composition of the battalion.

---

### **Step 1: Understand the Composition Requirement**
The battle demands a force composed of:
- **4 upper class soldiers**
- **8 lower class soldiers**

This means that **each battalion sent must be composed of exactly 4 upper class and 8 lower class soldiers**. The problem does not allow for any flexibility in this composition.

---

### **Step 2: Determine the Available Soldiers**
There are:
- **5 upper class soldiers**
- **10 lower class soldiers**

Thus, we are selecting 4 out of 5 upper class soldiers, and 8 out of 10 lower class soldiers.

---

### **Step 3: Use Combinations to Count Possible selections**

#### **Upper Class Soldiers**
We calculate how many ways we can choose 4 soldiers out of 5. This is a standard combination problem:

$$
\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5
$$

#### **Lower Class Soldiers**
Next, we calculate how many ways we can choose 8 out of 10 soldiers:

$$
\binom{10}{8} = \binom{10}{2} = \frac{10 \times 9}{2} = 45
$$

---

### **Step 4: Multiply the Two Combinations**
Since the choice of upper class and lower class soldiers is independent, the total number of distinct battalions that can be formed is the product of the two combinations:

$$
\text{Total Battalions} = \binom{5}{4} \times \binom{10}{8} = 5 \times 45 = 225
$$

---

### **Final Answer**
$$
\boxed{225}
$$

Solution: There are $\binom{5}{4}$ different ways to choose 4 from 5 upper class soldiers. For each of these, there are $\binom{10}{8}$ ways to choose 8 lower class soldiers. The number of different battalions, then, is $\binom{5}{4}\cdot \binom{10}{8} = \boxed{225}$.