Question

A particular baseball diamond is actually a square with 70​-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 square baseball diamond (which is shaped like a square), we can consider the problem using the Pythagorean theorem.

In a square diamond:

- Home plate is one corner of the square.
- First base is the adjacent corner, moving along one side of the square.
- Second base is the opposite corner, diagonally across from home plate.

Given that the sides of the diamond are 70 feet, we can use the coordinates for each of the bases:
- Home plate (H) at (0, 0)
- First base (F) at (70, 0)
- Second base (S) at (70, 70)

Now, to find the distance from home plate (H) to second base (S), we can use the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates for home plate (0, 0) and second base (70, 70):

\[
d = \sqrt{(70 - 0)^2 + (70 - 0)^2}
\]
\[
d = \sqrt{70^2 + 70^2}
\]
\[
d = \sqrt{4900 + 4900}
\]
\[
d = \sqrt{9800}
\]
\[
d = \sqrt{100 \cdot 98}
\]
\[
d = 10\sqrt{98}
\]

Next, we can simplify \(\sqrt{98}\):
\[
\sqrt{98} = \sqrt{49 \cdot 2} = 7\sqrt{2}
\]
Thus,
\[
d = 10 \cdot 7\sqrt{2} = 70\sqrt{2}
\]

Now, for the decimal approximation, we can calculate it using \(\sqrt{2} \approx 1.414\):
\[
d \approx 70 \cdot 1.414 \approx 99.01 \text{ feet}
\]

Therefore, the distance from home plate to second base in simplified radical form is \(70\sqrt{2}\) feet, and the decimal approximation is approximately \(99.01\) feet.