Question
Saul and Felipe are standing at the corner of a rectangular park. They will race to the opposite corner. The dimensions of the park are shown.
A diagram.Short description, A diagram.,Long description,
The diagram shows a rectangular park that is 400 feet long and 256 feet wide. A sidewalk follows the perimeter of the park. A path goes straight from the upper-left corner to the lower-right corner.
Saul runs at a constant rate of 8 feet per second along the sidewalk that surrounds the park. Felipe will walk at a constant rate along the path that goes straight from one corner to the other.
Question
How fast, in feet per second, must Felipe walk to finish the race at the same time as Saul? Round the answer to the nearest tenth. Enter the answer in the box.
Response area with 1 text input box
feet per second
A diagram.Short description, A diagram.,Long description,
The diagram shows a rectangular park that is 400 feet long and 256 feet wide. A sidewalk follows the perimeter of the park. A path goes straight from the upper-left corner to the lower-right corner.
Saul runs at a constant rate of 8 feet per second along the sidewalk that surrounds the park. Felipe will walk at a constant rate along the path that goes straight from one corner to the other.
Question
How fast, in feet per second, must Felipe walk to finish the race at the same time as Saul? Round the answer to the nearest tenth. Enter the answer in the box.
Response area with 1 text input box
feet per second
Answers
GPT-4o mini
To determine how fast Felipe must walk in order to finish the race at the same time as Saul, we first need to calculate the distances that each person will cover during the race.
1. **Calculate the distance Saul runs along the sidewalk:**
- The perimeter of the rectangular park can be calculated using the formula:
\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width})
\]
- Substituting the given dimensions:
\[
\text{Perimeter} = 2 \times (400 \text{ ft} + 256 \text{ ft}) = 2 \times 656 \text{ ft} = 1312 \text{ ft}
\]
However, Saul doesn't run the entire perimeter. He runs along the sidewalk from the upper-left corner to the lower-right corner, which follows two sides of the rectangle:
\[
\text{Distance Saul runs} = 400 \text{ ft} + 256 \text{ ft} = 656 \text{ ft}
\]
2. **Calculate the time Saul takes to complete the race:**
- Saul runs at a speed of 8 feet per second.
- The time taken by Saul to complete the 656 feet is:
\[
\text{Time (Saul)} = \frac{\text{Distance}}{\text{Speed}} = \frac{656 \text{ ft}}{8 \text{ ft/s}} = 82 \text{ seconds}
\]
3. **Calculate the distance Felipe walks along the diagonal path:**
- The diagonal distance can be calculated using the Pythagorean theorem:
\[
d = \sqrt{(\text{Length})^2 + (\text{Width})^2}
\]
- Substituting the given dimensions:
\[
d = \sqrt{(400 \text{ ft})^2 + (256 \text{ ft})^2} = \sqrt{160000 + 65536} = \sqrt{225536} \approx 474.2 \text{ ft}
\]
4. **Determine Felipe's required speed:**
- To finish at the same time as Saul (82 seconds), we need to determine the required walking speed for Felipe:
\[
\text{Speed (Felipe)} = \frac{\text{Distance}}{\text{Time}} = \frac{474.2 \text{ ft}}{82 \text{ seconds}} \approx 5.8 \text{ ft/s}
\]
So, Felipe must walk at approximately **5.8 feet per second** to finish the race at the same time as Saul.
The answer is:
\[
\boxed{5.8}
\]
1. **Calculate the distance Saul runs along the sidewalk:**
- The perimeter of the rectangular park can be calculated using the formula:
\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width})
\]
- Substituting the given dimensions:
\[
\text{Perimeter} = 2 \times (400 \text{ ft} + 256 \text{ ft}) = 2 \times 656 \text{ ft} = 1312 \text{ ft}
\]
However, Saul doesn't run the entire perimeter. He runs along the sidewalk from the upper-left corner to the lower-right corner, which follows two sides of the rectangle:
\[
\text{Distance Saul runs} = 400 \text{ ft} + 256 \text{ ft} = 656 \text{ ft}
\]
2. **Calculate the time Saul takes to complete the race:**
- Saul runs at a speed of 8 feet per second.
- The time taken by Saul to complete the 656 feet is:
\[
\text{Time (Saul)} = \frac{\text{Distance}}{\text{Speed}} = \frac{656 \text{ ft}}{8 \text{ ft/s}} = 82 \text{ seconds}
\]
3. **Calculate the distance Felipe walks along the diagonal path:**
- The diagonal distance can be calculated using the Pythagorean theorem:
\[
d = \sqrt{(\text{Length})^2 + (\text{Width})^2}
\]
- Substituting the given dimensions:
\[
d = \sqrt{(400 \text{ ft})^2 + (256 \text{ ft})^2} = \sqrt{160000 + 65536} = \sqrt{225536} \approx 474.2 \text{ ft}
\]
4. **Determine Felipe's required speed:**
- To finish at the same time as Saul (82 seconds), we need to determine the required walking speed for Felipe:
\[
\text{Speed (Felipe)} = \frac{\text{Distance}}{\text{Time}} = \frac{474.2 \text{ ft}}{82 \text{ seconds}} \approx 5.8 \text{ ft/s}
\]
So, Felipe must walk at approximately **5.8 feet per second** to finish the race at the same time as Saul.
The answer is:
\[
\boxed{5.8}
\]