Question
What is the distance a baseball player runs from home plate to third base on a square baseball diamond with 90ft sides at a constant speed of 20 feet per second?
Answers
GPT-4o mini
To find the distance a baseball player runs from home plate to third base on a square baseball diamond, we can first note the layout of the diamond. The distance from home plate to third base can be determined by using the Pythagorean theorem since the bases form a right triangle.
The distance from home plate (H) to first base (1B), and from first base to third base (3B) is each 90 feet. Therefore, we have two sides of a right triangle each measuring 90 feet.
Using the Pythagorean theorem:
\[
\text{Distance (D)} = \sqrt{(\text{side1})^2 + (\text{side2})^2}
\]
In this case, both sides are 90 feet:
\[
D = \sqrt{(90)^2 + (90)^2}
\]
\[
D = \sqrt{8100 + 8100}
\]
\[
D = \sqrt{16200}
\]
\[
D = 90\sqrt{2}
\]
Calculating \(90\sqrt{2}\):
\[
\sqrt{2} \approx 1.414
\]
\[
D \approx 90 \times 1.414 \approx 127.3 \text{ feet}
\]
Now, to find out how long it takes the player to run this distance at a constant speed of 20 feet per second, we use the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
Substituting in the values we found:
\[
\text{Time} = \frac{127.3 \text{ feet}}{20 \text{ feet/second}} \approx 6.365 \text{ seconds}
\]
So, the baseball player runs approximately 127.3 feet from home plate to third base at a constant speed of 20 feet per second, taking about 6.37 seconds to do so.
The distance from home plate (H) to first base (1B), and from first base to third base (3B) is each 90 feet. Therefore, we have two sides of a right triangle each measuring 90 feet.
Using the Pythagorean theorem:
\[
\text{Distance (D)} = \sqrt{(\text{side1})^2 + (\text{side2})^2}
\]
In this case, both sides are 90 feet:
\[
D = \sqrt{(90)^2 + (90)^2}
\]
\[
D = \sqrt{8100 + 8100}
\]
\[
D = \sqrt{16200}
\]
\[
D = 90\sqrt{2}
\]
Calculating \(90\sqrt{2}\):
\[
\sqrt{2} \approx 1.414
\]
\[
D \approx 90 \times 1.414 \approx 127.3 \text{ feet}
\]
Now, to find out how long it takes the player to run this distance at a constant speed of 20 feet per second, we use the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]
Substituting in the values we found:
\[
\text{Time} = \frac{127.3 \text{ feet}}{20 \text{ feet/second}} \approx 6.365 \text{ seconds}
\]
So, the baseball player runs approximately 127.3 feet from home plate to third base at a constant speed of 20 feet per second, taking about 6.37 seconds to do so.