Solve the problem.
-A person is watching a car from the top of a building. The car is traveling on a straight road away from the building. When first noticed, the angle of depression to the car is $48 ^ { \circ } 21 ^ { \prime }$. When the car stops, the angle of depression is $21 ^ { \circ } 43 ^ { \prime }$. The building is 250 feet tall. How far did the car travel from when it was first noticed until it stopped? Round to the nearest foot.
A) $384 \mathrm { ft }$
B) $432 \mathrm { ft }$
C) $605 \mathrm { ft }$
D) $405 \mathrm { ft }$

Question

Quizplus · Accepted Answer

To solve this problem, we use trigonometry, specifically the tangent of the angles of depression, which equals the opposite side (height of the building) over the adjacent side (distance from the building). Let $d_1$ be the distance from the building when the car was first noticed, and $d_2$ be the distance when the car stopped. The height of the building is 250 ft.For the first angle of depression, $48^\circ 21'$:$$ \tan(48^\circ 21') = \frac{250}{d_1} $$$$ d_1 = \frac{250}{\tan(48^\circ 21')} $$For the second angle of depression, $21^\circ 43'$:$$ \tan(21^\circ 43') = \frac{250}{d_2} $$$$ d_2 = \frac{250}{\tan(21^\circ 43')} $$The distance traveled by the car is $d_2 - d_1$. Plugging in the values:$$ d_1 = \frac{250}{\tan(48^\circ 21')} \approx 250 / 1.123 = 222.61 \, \text{ft} $$$$ d_2 = \frac{250}{\tan(21^\circ 43')} \approx 250 / 0.398 = 627.81 \, \text{ft} $$So, the distance traveled is $627.81 - 222.61 \approx 405$ ft, rounding to the nearest foot.

Solve the Problem 48∘21′48 ^ { \circ } 21 ^ { \prime }48∘21′

Solve the Problem $48 ^ { \circ } 21 ^ { \prime }$