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 find the angle between the side of the house and the cat's line of sight looking down at you, we can use some basic trigonometry.

Let's define the following:
- Let \( h \) be the height of the cat on the roof of the house.
- The horizontal distance from you to the base of the house is 10 feet.
- The diagonal distance from you to the cat is 18 feet.

We can form a right triangle where:
- One leg is the height of the house (\( h \)).
- The other leg is the distance from you to the base of the house (10 feet).
- The hypotenuse is the diagonal distance (18 feet).

Using the Pythagorean theorem, we can express this as:
\[
h^2 + 10^2 = 18^2
\]
Simplifying that:
\[
h^2 + 100 = 324
\]
Subtracting 100 from both sides gives:
\[
h^2 = 224
\]
Now, taking the square root of both sides:
\[
h = \sqrt{224} \approx 14.97 \quad \text{(approximately 15 feet)}
\]

Now that we have the height of the cat, we can find the angle \( \theta \) between the line of sight of the cat and the side of the house. The tangent of the angle \( \theta \) can be defined by:
\[
\tan(\theta) = \frac{h}{10}
\]
Substituting \( h \):
\[
\tan(\theta) = \frac{\sqrt{224}}{10} \approx \frac{14.97}{10} \approx 1.497
\]

Using the inverse tangent function to find the angle \( \theta \):
\[
\theta = \tan^{-1}(1.497)
\]

Calculating this:
\[
\theta \approx 56.31 \text{ degrees}
\]

Rounding to the nearest whole number:
\[
\theta \approx 56 \text{ degrees}
\]

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.