Question

A particular baseball diamond is actually a square with 87​-foot sides. What is the distance from home plate to second​ base? Express the answer in simplified radical form. Then find a decimal approximation.

Answer 1

To find the distance from home plate to second base on a baseball diamond that is a square with 87-foot sides, we can use the Pythagorean theorem.

Home plate and second base are two opposite corners of the square. Therefore, if we label the corners of the diamond as follows:
- Home plate (H)
- First base (F)
- Second base (S)
- Third base (T)

The distance \(d\) from home plate (H) to second base (S) can be calculated as the diagonal of the square formed by the bases.

Using the Pythagorean theorem, the diagonal \(d\) can be found using the formula:
\[
d = \sqrt{s^2 + s^2}
\]
where \(s\) is the length of a side of the square.

Given that \(s = 87\) feet, we plug in the numbers:
\[
d = \sqrt{87^2 + 87^2}
\]
\[
= \sqrt{2 \times 87^2}
\]
\[
= \sqrt{2} \times 87
\]

To express this in simplified radical form, we retain it as:
\[
d = 87\sqrt{2}
\]

Next, to approximate this value numerically, we can calculate \( \sqrt{2} \):
\[
\sqrt{2} \approx 1.414
\]
Therefore, we can calculate:
\[
d \approx 87 \times 1.414 \approx 123.018
\]

Thus, the distance from home plate to second base is:
- In simplified radical form: \( 87\sqrt{2} \) feet
- In decimal approximation: approximately \( 123.02 \) feet.