Question

question 5
1 pts
from the top of a 100 ft tall tower, bill sees a small fire to his left at an angle of depression of 55°. to the right bill sees a leaky oil barrel at an angle of depression of 40°. how far from the leaky oil barrel is the small fire? round your answer to the nearest tenth if necessary.

Explanation:

Step1: Find distance to fire

The angle of depression to the fire is $55^{\circ}$, and the height of the tower $h = 100$ ft. Let the horizontal - distance to the fire be $x_1$. Since $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, and the angle of depression is equal to the angle of elevation from the object to the top of the tower. $\tan55^{\circ}=\frac{100}{x_1}$, so $x_1=\frac{100}{\tan55^{\circ}}\approx\frac{100}{1.4281}\approx70.0$ ft.

Step2: Find distance to oil barrel

The angle of depression to the oil barrel is $40^{\circ}$, and the height of the tower $h = 100$ ft. Let the horizontal - distance to the oil barrel be $x_2$. Using $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\tan40^{\circ}=\frac{100}{x_2}$, so $x_2 = \frac{100}{\tan40^{\circ}}\approx\frac{100}{0.8391}\approx119.2$ ft.

Step3: Find distance between fire and oil barrel

Since the fire and the oil barrel are in opposite directions, the distance $d$ between them is $d=x_1 + x_2$. $d\approx70.0+158.6 = 228.6$ ft.

Answer:

228.6 ft