Problem 28 - Precalculus: Functions and Graphs

Question

A guy wire is attached to the top of a radio antenna and to a point on horizontal ground that is 40.0 meters from the base of the antenna. If the wire makes an angle of \(58^{\circ} 20^{\prime}\) with the ground, approximate the length of the wire.

Step-by-Step Solution

Answer

The length of the guy wire is approximately 75.90 meters.

Step 1: Understand the Problem

We are given a problem involving a guy wire attached to an antenna. We have a right triangle where one leg is the distance from the base to the point on the ground, which is 40 meters. The angle between the wire and the ground is given as 58° 20'. We need to find the hypotenuse, which represents the length of the wire.

Step 2: Convert the Angle to Decimal Form

First, convert the angle from degrees and minutes to decimal form for easier calculations. The angle is 58 degrees and 20 minutes. One minute is 1/60 of a degree, so we can calculate the decimal form: 58 degrees + \(\frac{20}{60}\) = 58.3333 degrees.

Step 3: Use the Right Triangle Definition with Cosine

The cosine of an angle in a right triangle is defined as the adjacent side over the hypotenuse. For our problem, the adjacent side is the ground distance (40 meters), and the hypotenuse is the wire length \(L\). The equation would be:\(\cos(58.3333^{\circ}) = \frac{40}{L}\)

Step 4: Solve for the Length of the Wire

Rearrange the equation to solve for \(L\):\(L = \frac{40}{\cos(58.3333^{\circ})}\)Calculate \(\cos(58.3333^{\circ})\) using a calculator. Then, divide 40 by this value to find the length of the wire.

Step 5: Calculate the Exact Length

Using a calculator, \( \cos(58.3333^{\circ}) \approx 0.5270 \). Substitute back into the equation:\( L = \frac{40}{0.5270} \approx 75.90 \text{ meters}\).Therefore, the length of the guy wire is approximately 75.90 meters.

Key Concepts

Right Triangle

Understanding the concept of a right triangle is crucial in solving our problem involving the guy wire and the radio antenna. A right triangle consists of three sides:

• The "adjacent side," which is next to the angle of interest.
• The "opposite side," which is across from the angle of interest.
• The "hypotenuse," which is the longest side and opposite the right angle.

In this scenario, consider the radio antenna as forming a right triangle with the ground. The base of the antenna to the ground point 40 meters away creates the adjacent side, and the angle between this side and the hypotenuse (the wire) is given as 58°20'. Knowing the relationships in a right triangle allows us to use trigonometric functions, like cosine, to find unknown side lengths.

Cosine Function

The cosine function offers a way to relate the adjacent side and the hypotenuse in a right triangle. Cosine, abbreviated as "cos," is a trigonometric function expressed as:

• Cosine of an angle = Adjacent side / Hypotenuse

For our problem, the adjacent side is the 40-meter horizontal distance, and the hypotenuse is the wire length. By setting up the equation: \[ \\cos(58.3333^{\circ}) = \frac{40}{L} \]We use the cosine function to express the relationship between the given angle, the known adjacent side, and the unknown hypotenuse.

Angle Conversion

Converting angles from degrees and minutes to decimal form makes calculations easier, especially while using trigonometric functions. Here's how you convert an angle like 58°20':

• Minutes are a fraction of a degree, where 60 minutes make up one degree.
• To convert 20 minutes to degrees, divide by 60: 20 / 60 = 0.3333.
• Add this to the 58 degrees: 58 + 0.3333 = 58.3333 degrees.

This conversion is essential because calculators typically require angles in decimal degrees for trigonometric calculations. Hence, in our exercise, the angle 58°20' is effectively 58.3333°.

Hypotenuse Calculation

Finding the hypotenuse involves rearranging the cosine equation to solve for the unknown wire length. Starting with our previously mentioned equation:\[L = \frac{40}{\cos(58.3333^{\circ})}\]Here's how to compute it step by step:

• Calculate \(\cos(58.3333^{\circ})\), which is approximately 0.5270.
• Substitute this value back into the equation: \(L = \frac{40}{0.5270}\).
• Perform the division to find \(L \approx 75.90\) meters.

This calculation confirms the guy wire's approximate length is 75.90 meters, by using the relationship between the sides of a right triangle and the angle's cosine.