Problem 29 - Precalculus: A Concise Course

Question

A cellular telephone tower that is 150 feet tall is placed on top of a mountain that is 1200 feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet above sea level?

Step-by-Step Solution

Answer

The angle of depression from the top of the tower to the cell phone user is the value of \( \Theta \) rounded to two decimal places \( = \text{arctan}(\frac{950}{26400})\).

Step 1: Preparing the Problem

First, draw a diagram of the situation, representing the tower, the mountain, and the cell phone user. As the mountain and the tower heights are given relative to sea level, calculate the difference in height between the top of the tower and the cell phone user: \(150 + 1200 - 400 = 950\) feet. Note that the horizontal distance to the person is given in miles. Convert this to feet using the fact that 1 mile is approximately 5280 feet: \(5 * 5280 = 26400\) feet.

Step 2: Apply Tangent

In the right triangle formed, the angle of depression can be determined using the tangent of the angle, which is the ratio of the opposite side (difference in height) to the adjacent side (horizontal distance). Therefore, use the formula: \(\text{Tan}(\Theta) = \frac{950}{26400}\).

Step 3: Finding the angle of depression

In order to find the value of \( \Theta \), take the inverse tangent (or arctangent) of the value found in Step 2. This is done using a scientific calculator. Find \(\Theta = \text{arctan}(\frac{950}{26400})\).