Problem 59 - Precalculus Essentials

Question

A telephone pole is 60 feet tall. A guy wire 75 feet long is attached from the ground to the top of the pole. Find the angle between the wire and the pole to the nearest degree. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The angle between the pole and the wire, to the nearest degree, is approximately 41 degrees.

Step 1: Identify the knowns

In the right triangle formed by the pole, the wire, and the ground, the pole is the adjacent side (60 feet), the wire is the hypotenuse (75 feet), and the angle between the wire and the pole is the unknown. Let's denote the angle by \( θ \).

Step 2: Set up the equation

By definition, cosine of an angle is the ratio of the adjacent side to the hypotenuse. In this case, \( cos(θ) = \frac{60}{75} \).

Step 3: Solve for \( θ \)

To find \( θ \), use the inverse cosine (or arccos). So, \( θ = arccos(\frac{60}{75}) \).

Step 4: Convert to degrees

Since the question asks for the angle to the nearest degree, we need to convert \( θ \) from radians to degrees by multiplying by \( \frac{180}{π} \).

Step 5: Round off to nearest degree

Finally, round off \( θ \) to the nearest degree as requested in the problem statement.