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Chapter 10: Problem 46

A police helicopter is flying at 800 feet. A stolen car is sighted at an angle of depression of \(72^{\circ}\). Find the distance of the stolen car, to the nearest foot, from a point directly below the helicopter.

Short Answer

Expert verified
The distance of the stolen car from a point directly below the helicopter is approximately 249 feet.

Step by step solution

01

Identifying Given Information

The height at which the helicopter is flying is 800 feet. The angle of depression, which is the same as the angle of elevation from the ground to the helicopter, is \(72^{\circ}\). The distance of the stolen car from a point directly below the helicopter is what we need to find.
02

Applying Trigonometric Ratio(Tangent)

In a right triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. Symbolically, \(tan(胃) = \frac{opposite}{adjacent}\). Here, 胃 is the angle of elevation, the opposite side is the height of the helicopter, and the adjacent side is the distance of the car from the point direct below the helicopter, which we're trying to find.
03

Solving for the unknown (adjacent)

Rearranging the equation to solve for the adjacent side, we get \(adjacent = \frac{opposite}{tan(胃)}\). Substituting the given values in, we have \(adjacent = \frac{800}{tan(72)}\) feet.
04

Calculating the Result

Using a calculator, calculate the value of \(adjacent = \frac{800}{tan(72)}\) to find out the distance.

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