Question

n Campbell, the library is due south of the courthouse and due west of the community swimming pool. If the distance between the library and the courthouse is 24 kilometers and the distance between the courthouse and the city pool is 30 kilometers, how far is the library from the community pool?

Answer 1

We can visualize the positions of the library (L), courthouse (C), and community swimming pool (P) as a right triangle based on the provided directions.

1. The library is due south of the courthouse, which means if we place the courthouse at the coordinates (0, 0), the library will be directly south, at (0, -24) because the distance between the library and courthouse is 24 kilometers.

2. The community swimming pool is due east of the courthouse, which places it at (30, 0) because the distance from the courthouse to the community pool is 30 kilometers.

Now we have the three points:
- Courthouse (C) at (0, 0)
- Library (L) at (0, -24)
- Community Pool (P) at (30, 0)

To find the distance between the library and the community pool, we can apply the distance formula, which is:

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

Substituting the coordinates of the library (0, -24) and the community pool (30, 0):

\[
d = \sqrt{(30 - 0)^2 + (0 - (-24))^2}
\]
\[
d = \sqrt{30^2 + 24^2}
\]
\[
= \sqrt{900 + 576}
\]
\[
= \sqrt{1476}
\]
\[
= \sqrt{4 \times 369}
\]
\[
= 2\sqrt{369}
\]

Calculating \(\sqrt{369}\) approximately:
\(\sqrt{369} \approx 19.2\)

Thus,

\[
d \approx 2 \times 19.2 = 38.4
\]

So, the distance between the library and the community pool is approximately \(38.4\) kilometers.