Problem 36 - Algebra and Trigonometry

Question

Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 \(\mathrm{ft}\) and 180 \(\mathrm{ft}\) long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?

Step-by-Step Solution

Answer

The anchor points are approximately 166.87 ft apart.

Step 1: Understand the Problem Geometry

We have a right triangle formed by the shorter wire, the ground, and the antenna. The hypotenuse is the length of the shorter wire (165 ft), the angle with the ground is given as 67°, and we need to find the distance between the anchor points on either side of the antenna, which is the sum of two horizontal distances below each wire.

Step 2: Calculate the Height of the Antenna

The height of the antenna is the vertical side of the right triangle. Use the sine function: \( \sin(67^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{165}\). Thus, \( h = 165 \times \sin(67^\circ) \approx 151.53 \text{ ft} \).

Step 3: Find the Horizontal Distance for the Shorter Wire

Using the cosine function for the shorter wire's triangle: \( \cos(67^\circ) = \frac{\text{adjacent}}{165} = \frac{d_1}{165} \). Solving for \(d_1\), \( d_1 = 165 \times \cos(67^\circ) \approx 64.43 \text{ ft} \).

Step 4: Calculate the Horizontal Distance for the Longer Wire

For the longer wire, which also reaches the antenna height, use the height and the Pythagorean Theorem: \( d_2 = \sqrt{180^2 - h^2} = \sqrt{180^2 - 151.53^2} \approx 102.44 \text{ ft} \).

Step 5: Determine the Total Distance Between Anchor Points

The total distance between the anchor points is the sum of the horizontal distances on opposite sides of the antenna: \( d_1 + d_2 = 64.43 + 102.44 \approx 166.87 \text{ ft} \).

Key Concepts

Right Triangle

A right triangle is a special kind of triangle that has one angle equal to 90 degrees. This makes it very useful in trigonometry because it allows us to apply specific functions and theorems, like sine, cosine, and the Pythagorean theorem.
In our exercise, the shorter wire, the ground, and the antenna form a right triangle. Here, the hypotenuse is the side opposite the right angle, and it measures 165 feet, which is the length of the shorter wire. The right angle is formed where the antenna meets the ground. This structure makes it easier for us to use trigonometric functions to find unknown lengths in the triangle.

Sine Function

The sine function is a trigonometric function that relates a non-right angle in a right triangle to the ratio of the opposite side over the hypotenuse.
The formula is:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

In our problem, we use the sine function to find the height of the antenna. We know the hypotenuse (165 ft) and the angle with the ground (67°). So, the height \( h \) is the opposite side when using sine:

• \( h = 165 \times \sin(67°) \approx 151.53 \text{ ft} \)

This calculation gives us the vertical distance from the top of the antenna to the ground.

Cosine Function

The cosine function relates the adjacent side of a non-right angle in a right triangle to the hypotenuse.
The formula for cosine is:

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

In the context of our exercise, we use the cosine function to determine the horizontal distance from the base of the antenna to the anchor point of the shorter wire. Given that the hypotenuse is 165 ft and the angle with the ground is 67°, we can find this horizontal distance:

• \( d_1 = 165 \times \cos(67°) \approx 64.43 \text{ ft} \)

This horizontal distance is one part needed to find the total distance between the anchor points.

Pythagorean Theorem

The Pythagorean theorem is crucial in understanding and solving problems involving right triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The formula is:

• \( a^2 + b^2 = c^2 \)

Here, the longer wire has a length of 180 feet, which serves as the hypotenuse for its own right triangle formed with the ground and the antenna's height. We already calculated the height of the antenna as \( h = 151.53 \text{ ft} \). Using this, we find the second horizontal distance \( d_2 \) using:

• \( d_2 = \sqrt{180^2 - 151.53^2} \approx 102.44 \text{ ft} \)

Combining this with the \( d_1 \) we found using the cosine function, we get the total separation of the anchor points as approximately 166.87 feet, showcasing how each tool in trigonometry can be applied effectively.