A man is in a tree house 10 ft. above the ground. He is looking at the top of another tree that is 22 ft. tall. The bases of the trees are 30 ft. apart. What is the angle of elevation from the man’s feet to the top of the tree? Round to the nearest degree.

Question

(1 point)
Responses

68°
68°

66°
66°

24°
24°

22°

Answer 1

To find the angle of elevation from the man's feet to the top of the other tree, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.

1. Determine the height difference (opposite side):

   • The height of the other tree is 22 ft.
   • The height of the man in the tree house is 10 ft.
   • Therefore, the height difference is: \[ 22 \text{ ft} - 10 \text{ ft} = 12 \text{ ft} \]

2. Determine the distance between the bases of the trees (adjacent side):

   • The base of the trees is 30 ft apart.

3. Use the tangent function:

   • The angle of elevation \( \theta \) can be calculated using: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{30} \]

4. Calculate \( \tan(\theta) \): \[ \tan(\theta) = \frac{12}{30} = \frac{2}{5} = 0.4 \]

5. Find the angle \( \theta \):

   • To find \( \theta \), we take the arctangent (inverse tangent) of 0.4: \[ \theta = \tan^{-1}(0.4) \]

6. Calculate the angle:

   • Using a calculator, we find: \[ \theta \approx 22° \]

Thus, the angle of elevation from the man's feet to the top of the tree is 22°.