Solve the problem.
-A person is watching a car from the top of a building. The car is traveling on a straight road directly toward the building. When first noticed the angle of depression to the car is $29 ^ { \circ } 46 ^ { \prime }$. When the car stops, the angle of depression is $41 ^ { \circ } 17 ^ { \prime }$. The building is 230 feet tall. How far did the car travel from when it was first noticed until it stopped? Round your answer to the hundredths place.
A) $140.19 \mathrm { ft }$
B) $340.29 \mathrm { ft }$
C) $166.4 \mathrm { ft }$
D) $119.12 \mathrm { ft }$

Question

Quizplus · Accepted Answer

To solve this problem, we use trigonometry, specifically the tangent of the angles of depression, which gives us the ratio of the opposite side (distance traveled by the car) to the adjacent side (height of the building). Let $d_1$ and $d_2$ be the distances of the car from the building when it was first noticed and when it stopped, respectively.1. For the angle of depression of $29^\circ 46'$, we have:$$ \tan(29^\circ 46') = \frac{d_1}{230} $$Solving for $d_1$ gives us the distance from the building when the car was first noticed.2. For the angle of depression of $41^\circ 17'$, we have:$$ \tan(41^\circ 17') = \frac{d_2}{230} $$Solving for $d_2$ gives us the distance from the building when the car stopped.The distance traveled by the car is the difference between these two distances, $d_1 - d_2$.Using the given angles and the height of the building (230 ft), we calculate $d_1$ and $d_2$, and then find the difference to get the distance traveled by the car. After performing the calculations, the result is approximately $140.19$ feet, which corresponds to choice A.

Solve the Problem 29∘46′29 ^ { \circ } 46 ^ { \prime }29∘46′

Solve the Problem $29 ^ { \circ } 46 ^ { \prime }$