Question

Sam looks out the window of a building that is 60 feet above the ground. He watches a car travel

away from the building along a straight road. When he first sees the car, his angle of depression to th

car is 15 degrees. He watches the car travel until his angle of depression to the car is 4 degrees.


How far did the car travel while Sam was watching?


A) 223.92 feet

B) 634.12 feet

C) 858.04 feet

D)1081.96 feet

Answer 1

B) 634.12 feet

Explanation:

What is the Angle of Depression?

The angle of depression is defined as an angle constructed by a horizontal line and the line joining the object and the observer's eye.

In the given situation, the first angle of depression is 15° and the second angle of depression is 4°.

The attached image models this situation

AB represents the building

BC is the line of sight from the top of the building to the first location of the car at 15° angle of depression; AC is the distance of the first sighting of the car from the base of the building

BD represents the line of sight at the second position of the car at 4° angle of depression; AD is the distance of the second sighting of the car from the base of the building

The distance covered by the car while Sam was watching = AD - AC

Consider the right triangle ABC.
m∠ABC = 90 - 15° = 75°

Using the formula for the tangent of an angle in a right triangle,

`\tan 75^\circ = (AC)/(AB) = (AC)/(60) \\\\(AC)/(60) =\tan 75^\circ\\\\AC = 60 \cdot \tan 75^\circ\\\\AC = 223.92\: feet`

Now consider the right triangle ABD

AD represents the distance of the car from base of the building at second sighting.

m∠ABD = 90 - m∠DBF = 90 - 4 = 86°

Using the formula for the tangent of an angle:

`\tan(86^\circ) = (AD)/(AB)\\\\(AD)/(AB) = \tan(86^\circ)\\\\AD = AB \cdot \tan(86^\circ)\\\\AD = 60 \cdot \tan(86^\circ)\\\\AD = 858.04\:feet`

The distance traveled by the car between the two sightings is
AD - AC

= 858.04 - 223.92 = 634.12 feet

This would be option B)