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.
Answers
GPT-4o mini
To find the distance saved by the bird flying directly from point A to point B instead of the car's route, we first need to understand the scenario.
1. The square has sides of 5 miles. If we consider point A at the top-left corner of the square (let's say coordinates (0, 5)) and point B at the bottom-right corner (coordinates (5, 0)), the car will take a two-part journey: it will go south 5 miles and then east 5 miles.
2. The car's distance traveled is:
\[
\text{Distance by car} = 5 \text{ miles (south)} + 5 \text{ miles (east)} = 10 \text{ miles}
\]
3. The bird, however, would fly diagonally from point A to point B. The straight-line distance can be calculated using the Pythagorean theorem:
\[
\text{Distance by bird} = \sqrt{(5 - 0)^2 + (0 - 5)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
\]
4. We can compute \( 5\sqrt{2} \) to find its decimal value.
- Approximating \( \sqrt{2} \approx 1.414 \):
\[
5\sqrt{2} \approx 5 \times 1.414 \approx 7.07 \text{ miles}
\]
5. Now, we calculate the distance saved by the bird:
\[
\text{Distance saved} = \text{Distance by car} - \text{Distance by bird}
\]
\[
\text{Distance saved} = 10 \text{ miles} - 7.07 \text{ miles} \approx 2.93 \text{ miles}
\]
6. Rounding this to the nearest tenth, we find:
\[
\text{Distance saved} \approx 2.9 \text{ miles}
\]
Thus, the distance saved by the bird is approximately **2.9 miles**.
1. The square has sides of 5 miles. If we consider point A at the top-left corner of the square (let's say coordinates (0, 5)) and point B at the bottom-right corner (coordinates (5, 0)), the car will take a two-part journey: it will go south 5 miles and then east 5 miles.
2. The car's distance traveled is:
\[
\text{Distance by car} = 5 \text{ miles (south)} + 5 \text{ miles (east)} = 10 \text{ miles}
\]
3. The bird, however, would fly diagonally from point A to point B. The straight-line distance can be calculated using the Pythagorean theorem:
\[
\text{Distance by bird} = \sqrt{(5 - 0)^2 + (0 - 5)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
\]
4. We can compute \( 5\sqrt{2} \) to find its decimal value.
- Approximating \( \sqrt{2} \approx 1.414 \):
\[
5\sqrt{2} \approx 5 \times 1.414 \approx 7.07 \text{ miles}
\]
5. Now, we calculate the distance saved by the bird:
\[
\text{Distance saved} = \text{Distance by car} - \text{Distance by bird}
\]
\[
\text{Distance saved} = 10 \text{ miles} - 7.07 \text{ miles} \approx 2.93 \text{ miles}
\]
6. Rounding this to the nearest tenth, we find:
\[
\text{Distance saved} \approx 2.9 \text{ miles}
\]
Thus, the distance saved by the bird is approximately **2.9 miles**.