Problem 30 - A Graphical Approach to Precalculus with Limits

Question

Length of a Guy Wire A weather tower used to measure wind speed has a guy wire attached to it 175 feet above the ground. The angle between the wire and the vertical tower is \(57.0^{\circ} .\) Approximate the length of the guy wire.

Step-by-Step Solution

Answer

The length of the guy wire is approximately 208.66 feet.

Step 1: Identify the right triangle components

In this problem, the weather tower, the guy wire, and the ground form a right triangle. The tower acts as one side of the triangle (the opposite), the ground is the adjacent side, and the guy wire is the hypotenuse. The angle given above the ground is the angle between the guy wire and the tower, which is 57 degrees.

Step 2: Apply the trigonometric function

To find the length of the guy wire, use the sine function of trigonometry, which relates the opposite side and the hypotenuse.The sine function is defined as:\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}.\]Here, \(\theta = 57^{\circ}\) and the opposite side is 175 feet.

Step 3: Solve for the length of the hypotenuse

Rearrange the sine function to solve for the hypotenuse:\[\text{hypotenuse} = \frac{\text{opposite side}}{\sin(\theta)} = \frac{175}{\sin(57^{\circ})}. \]

Step 4: Perform the calculation

Using a calculator, find \(\sin(57^{\circ})\), and then compute the length of the hypotenuse:\( \sin(57^{\circ}) \approx 0.8387 \).Now, calculate:\(\text{hypotenuse} = \frac{175}{0.8387} \approx 208.66 \text{ feet}.\)

Key Concepts

Right Triangle

When we talk about right triangles, these are triangles that contain one angle measuring exactly 90 degrees. This particular angle forms a perfect square corner, making it easy to identify.
In the context of our exercise, the right triangle is formed by:

• The vertical weather tower, representing one side.
• The ground acting as the horizontal base, another side.
• The guy wire, which completes the triangle as the hypotenuse (the longest side in a right triangle).

Right triangles play a crucial role in trigonometry, as they allow us to use special functions to find unknown sides or angles. By identifying the parts of a right triangle, we can apply these trigonometric principles effectively.

Sine Function

The sine function is a fundamental part of trigonometry. It establishes a relationship between an angle in a right triangle and the lengths of its sides.
Simply put, the sine of an angle is determined by the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, this is expressed as:\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\]For our exercise, the angle \(\theta\) provided is 57 degrees, and the opposite side, which is the height of the tower, measures 175 feet. By plugging these into the sine formula, we can solve for the hypotenuse, which is the length of the guy wire.Understanding how the sine function works can help solve various real-world problems that involve indirect measurements. This makes it a powerful tool in both academic and practical applications.

Angle Measurement

Angles in geometry are fundamental units of measure that allow us to describe the orientation and inclination of lines and shapes. In our problem, the angle of 57 degrees is specified between the guy wire and the vertical tower. This angle tells us how steeply the wire ascends from the tower to the ground.
When dealing with angles, it’s essential to understand the tools needed to measure them, including degrees, which is the unit used here. Each complete circle contains 360 degrees, and understanding this can help in visualizing orientations.
In practical applications, especially with right triangles, knowing the measure of one angle can provide valuable information about the relationships between the sides. This allows us to utilize trigonometric functions like sine, which in turn helps in solving for unknown side lengths, just like we did in calculating the guy wire's length.