How far apart are the cars? (Round your answer to the nearest foot

Question

An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is 36° and to the other is 56°. How far apart are the cars? (Round your answer to the nearest foot.)

Answer 1

The angle of depression is the angle you'd have to move your eyes downwards to look at the car from the plane. Let's start with the first car, with the 36° angle of depression. Draw an upside-down right triangle with vertices at the plane, the car, and the point 5150 feet in the air above the car (level with the plane). The vertex at the plane is 36° and the right angle is the vertex in the air above the car. The length of the leg from the car to the point in the air above the car is 5150 feet. We'd like to find the length of the leg from the plane to the point in the air above the car. Since the two sides involved are the legs of the triangle, use tangent:

 

tan = opp/adj

tan(36°) = 5150/x

0.727 = 5150/x

0.727x = 5150

x = 7088.37

 

That means the first car is 7088.37 feet from the point on the highway below the plane.

 

We can do something similar with the second car, which has an angle of depression of 56° from the plane. Again, the leg from the car to the point in the air above the car (level with the plane) is 5150 feet, the right angle is at the vertex at the point in the air above the car, and the 56° angle is at the vertex at the plane. We're looking for the length of the other leg, which runs from the plane to the point in the air above the car. Use tangent:

 

tan(56°) = 5150/x

1.483 = 5150/x

1.483x = 5150

x = 3473.72

 

That means the second car is 3473.72 feet from the point on the highway below the plane.

 

 

Add the two distances together to get the total distance from car to car:

 

7088.37 + 3473.72 = 10562.09

 

So, rounded to the nearest foot, the cars are 10,562 feet apart.