Problem 46 - Algebra and Trigonometry

Question

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.

Step-by-Step Solution

Answer

The distance of the stolen car from a point directly below the helicopter is approximately 257 feet, to the nearest foot.

Step 1: Understand the Problem

As we are given the height of the helicopter and the angle of depression, we need to set up a right triangle and use the properties of trigonometry to find the distance of the stolen car from the helicopter. The question requires us to find the distance from a point directly below the helicopter (which will serve as our adjacent side in our right triangle). The angle of depression is \(72^{\circ}. \)

Step 2: Apply the Trigonometric Rule

In this case, we use the tangent of the angle (which is defined as the ratio of the opposite side over the adjacent side) to set up our equation. The tangent of \(72^{\circ}\) is equal to the height of the helicopter divided by the distance to the car (our unknown). In mathematical terms: \(tan(72^{\circ}) = \frac{800}{x}\).

Step 3: Solve for the Unknown

To solve for \(x\) (the distance to the car), we'll rearrange the equation to isolate \(x\), which gives us \(x = \frac{800}{tan(72^{\circ})}\). Using a calculator to find the value of \(tan(72^{\circ})\) we can then substitute to find the value of \(x\).