Question

A cat climbs on top of a house and looks down at you. You are standing 10 feet away from the house and there is an 18-foot diagonal distance between you and the cat. Find the angle between the side of the house and the cat's line of sight looking down at you. Round your answer to the nearest whole number.

Answer 1

To solve for the angle between the side of the house and the cat's line of sight, we can use trigonometry. We are given the following:

- The distance from you to the house (horizontal distance) is \(10\) feet.
- The diagonal distance from the cat to you is \(18\) feet.

We can denote the angle we want to find as \(\theta\). According to the definition of the cosine function in a right triangle, we have:

\[
\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]

Here, the adjacent side is the horizontal distance from you to the base of the house (which is \(10\) feet), and the hypotenuse is the diagonal distance (which is \(18\) feet). Therefore, we can write:

\[
\cos(\theta) = \frac{10}{18}
\]

To find \(\theta\), we first calculate \(\frac{10}{18}\):

\[
\frac{10}{18} = \frac{5}{9} \approx 0.5556
\]

Next, we take the arccosine of this value:

\[
\theta = \cos^{-1}\left(\frac{5}{9}\right)
\]

Using a calculator, we find:

\[
\theta \approx 56.31^\circ
\]

Rounding this to the nearest whole number, we get:

\[
\theta \approx 56^\circ
\]

Thus, the angle between the side of the house and the cat's line of sight looking down at you is approximately \( \boxed{56} \) degrees.