Question
Each side of a square is 5 miles long. How much distance is saved by a bird that would fly directly from point A to point B instead of a car that took a direct route south and then east? Round your answer to the nearest tenth of a mile.
Show all work and reasoning.
Show all work and reasoning.
Answers
GPT-4o mini
Let's break down the problem step-by-step.
1. **Understanding the layout**: We have a square with each side measuring 5 miles. We label the corners of the square as follows:
- Point A at the top-left corner (coordinates (0, 5))
- Point B at the bottom-right corner (coordinates (5, 0))
2. **Distance the car travels**: The car travels directly south and then directly east to reach Point B:
- First, it travels south from Point A (0, 5) to (0, 0). This distance is:
\[
5 \text{ miles} \, (\text{from 5 to 0 on the y-axis})
\]
- Then, it travels east from (0, 0) to (5, 0). This distance is:
\[
5 \text{ miles} \, (\text{from 0 to 5 on the x-axis})
\]
- Therefore, the total distance traveled by the car is:
\[
5 \text{ miles} + 5 \text{ miles} = 10 \text{ miles}
\]
3. **Distance the bird travels**: The bird flies directly from Point A to Point B in a straight line. To find this distance, we can use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Here:
- For Point A \((x_1, y_1) = (0, 5)\) and Point B \((x_2, y_2) = (5, 0)\), we plug in the coordinates:
\[
d = \sqrt{(5 - 0)^2 + (0 - 5)^2}
\]
- Calculating inside the square root:
\[
d = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50}
\]
- Simplifying \(\sqrt{50}\):
\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\]
- Using \( \sqrt{2} \approx 1.414\):
\[
d \approx 5 \times 1.414 \approx 7.07 \text{ miles}
\]
4. **Calculating the distance saved**: The distance saved is the difference between the distance the car travels and the distance the bird travels:
\[
\text{Distance saved} = \text{Distance by car} - \text{Distance by bird}
\]
\[
\text{Distance saved} = 10 \text{ miles} - 7.07 \text{ miles} \approx 2.93 \text{ miles}
\]
5. **Rounding to the nearest tenth**:
\[
\text{Distance saved} \approx 2.9 \text{ miles}
\]
Thus, the distance saved by the bird is approximately **2.9 miles**.
1. **Understanding the layout**: We have a square with each side measuring 5 miles. We label the corners of the square as follows:
- Point A at the top-left corner (coordinates (0, 5))
- Point B at the bottom-right corner (coordinates (5, 0))
2. **Distance the car travels**: The car travels directly south and then directly east to reach Point B:
- First, it travels south from Point A (0, 5) to (0, 0). This distance is:
\[
5 \text{ miles} \, (\text{from 5 to 0 on the y-axis})
\]
- Then, it travels east from (0, 0) to (5, 0). This distance is:
\[
5 \text{ miles} \, (\text{from 0 to 5 on the x-axis})
\]
- Therefore, the total distance traveled by the car is:
\[
5 \text{ miles} + 5 \text{ miles} = 10 \text{ miles}
\]
3. **Distance the bird travels**: The bird flies directly from Point A to Point B in a straight line. To find this distance, we can use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Here:
- For Point A \((x_1, y_1) = (0, 5)\) and Point B \((x_2, y_2) = (5, 0)\), we plug in the coordinates:
\[
d = \sqrt{(5 - 0)^2 + (0 - 5)^2}
\]
- Calculating inside the square root:
\[
d = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50}
\]
- Simplifying \(\sqrt{50}\):
\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\]
- Using \( \sqrt{2} \approx 1.414\):
\[
d \approx 5 \times 1.414 \approx 7.07 \text{ miles}
\]
4. **Calculating the distance saved**: The distance saved is the difference between the distance the car travels and the distance the bird travels:
\[
\text{Distance saved} = \text{Distance by car} - \text{Distance by bird}
\]
\[
\text{Distance saved} = 10 \text{ miles} - 7.07 \text{ miles} \approx 2.93 \text{ miles}
\]
5. **Rounding to the nearest tenth**:
\[
\text{Distance saved} \approx 2.9 \text{ miles}
\]
Thus, the distance saved by the bird is approximately **2.9 miles**.