Question

A cat climbs onto a roof and looks down at you. You are standing 12 feet away from the house. There is an 18 foot diagonal distance between you and the cat. What is the angle the cat is looking down at you? Round to the nearest whole degree. (7 points)

Answer 1

To find the angle at which the cat is looking down at you, we can use the concept of right triangles and trigonometric functions.

1. **Identify the sides of the triangle**:
- The horizontal distance from you to the house (adjacent side) = 12 feet.
- The diagonal distance from you to the cat (hypotenuse) = 18 feet.
- We will use the Pythagorean theorem to find the height of the cat (opposite side).

2. **Use the Pythagorean theorem**:
\[
c^2 = a^2 + b^2
\]
Where
- \( c \) is the hypotenuse (18 feet),
- \( a \) is the adjacent side (12 feet),
- \( b \) is the opposite side (height of the cat).

Rearranging the formula to solve for \( b \):
\[
b^2 = c^2 - a^2
\]
\[
b^2 = 18^2 - 12^2
\]
\[
b^2 = 324 - 144
\]
\[
b^2 = 180
\]
\[
b = \sqrt{180} \approx 13.42 \text{ feet}
\]

3. **Calculate the angle**:
Now that we have the height of the cat, we can use the tangent function to find the angle \( \theta \):
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a} = \frac{13.42}{12}
\]

\[
\tan(\theta) \approx 1.1183
\]

4. **Find the angle using arctan**:
To find \( \theta \):
\[
\theta = \tan^{-1}(1.1183) \approx 48.37^\circ
\]

5. **Round to the nearest whole degree**:
So, rounding \( 48.37^\circ \) gives us \( 48^\circ \).

Thus, the angle the cat is looking down at you is approximately **48 degrees**.