Problem 27 - Precalculus: A Concise Course

Question

An engineer erects a 75 -foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground 50 feet from its base.

Step-by-Step Solution

Answer

The angle of elevation to the top of the tower from a point 50 feet from its base on level ground is approximately \( \tan^{-1}(1.5) \) which is roughly 56.31 degrees.

Step 1: Understand the problem

An engineer has erected a 75-foot cellular telephone tower. They are now attempting to calculate the angle of elevation to the top of the tower from a point that is 50 feet from its base on level ground. This forms a right-angled triangle, and the objective is to determine one of the angles.

Step 2: Apply the tangent function

In a right triangle, the tangent of an angle is equivalent to the ratio of the length of the side opposite to the angle (in this case, the height of the tower), to the length of the side adjacent to the angle (in this case, the distance from the base of the tower). Thus, the tangent of the angle of elevation (\( \theta \)) is given by \( \tan(\theta) = \frac{75}{50} = 1.5 \).

Step 3: Calculate the angle of elevation

To get the angle of elevation, we take the inverse tangent of 1.5. This can be done using a scientific calculator and should result in \( \theta = \tan^{-1}(1.5) \).