Problem 26 - Algebra 2 and Trigonometry

Question

A vertical telephone pole that is 15 feet high is braced by two wires from the top of the pole to two points on the ground that are 5.0 feet apart on the same side of the pole and in a straight line with the foot of the pole. The shorter wire makes an angle of 65 degrees with the ground. Find the length of each wire to the nearest tenth.

Step-by-Step Solution

Answer

The shorter wire is approximately 16.5 feet long, and the longer wire is 19.2 feet long.

Step 1: Understand the Problem

We need to determine the lengths of two wires that brace a pole. The first wire makes a 65-degree angle with the ground, and the second wire is attached to a point on the ground 5 feet away from the first contact point. The pole height is 15 feet.

Step 2: Use Trigonometric Function for Shorter Wire

To find the length of the shorter wire, use the sine function: \( \sin(\theta) = \frac{opposite}{hypotenuse} \). Here, the opposite side is the pole's height (15 feet), and \( \theta = 65^\circ \). We have \( \sin(65^\circ) = \frac{15}{L_1} \). Solve for \( L_1 \): \( L_1 = \frac{15}{\sin(65^\circ)} \).

Step 3: Compute Length of Shorter Wire

Calculate \( L_1 = \frac{15}{\sin(65^\circ)} \). Using a calculator, find \( \sin(65^\circ) \approx 0.9063 \), so \( L_1 \approx \frac{15}{0.9063} \approx 16.5 \) feet.

Step 4: Use Trigonometric Function for Longer Wire

The longer wire reaches 5 feet further along the ground. Since the ground distance for the shorter wire (distance from the pole to the point of the first wire's contact with the ground) is \( 15 / \tan(65^\circ) \), find this distance first, and add 5 feet: \( x = \frac{15}{\tan(65^\circ)} + 5 \).

Step 5: Compute Ground Distance of Longer Wire

Calculate \( \tan(65^\circ) \approx 2.1445 \), giving \( \frac{15}{2.1445} \approx 7.0 \) feet. The total ground distance for the longer wire is \( 7.0 + 5 = 12.0 \) feet.

Step 6: Calculate Length of Longer Wire

Use the Pythagorean theorem for the longer wire: \( L_2 = \sqrt{15^2 + 12^2} \). Compute \( L_2 \approx \sqrt{225 + 144} = \sqrt{369} \approx 19.2 \) feet.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental tools in mathematics, especially when dealing with triangles. These functions include sine, cosine, and tangent. They relate the angles of a triangle to the lengths of its sides. In our exercise, they help us find the lengths of the wires bracing the telephone pole.

• The sine function relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle.
• The cosine function connects the angle to the adjacent side over the hypotenuse.
• Lastly, the tangent function relates an angle to the opposite side over the adjacent side.

By using these functions, it becomes easier to solve problems involving right triangles, like the one with the telephone pole.
In this problem, we primarily use the sine and tangent functions to determine the necessary distances and angles.

Sine Function

The sine function is key to solving the shorter wire length in the exercise. To understand how this function works, think of a right triangle. The sine of an angle is calculated as the ratio of the length of the side opposite the angle to the hypotenuse.
For example, in the problem you have a right triangle formed by the pole, the ground, and the shorter wire. We know:

• Height of pole = 15 feet
• Angle with ground = 65 degrees
• Wire length (hypotenuse) = ? (this is what we need to find)

The formula for sine is:\[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]
Applying this to our problem gives us:
\[\sin(65^\circ) = \frac{15}{L_1}\]
Solving for \(L_1\) (the wire length) involves a simple rearrangement:
\[L_1 = \frac{15}{\sin(65^\circ)}\]
This mathematical relationship allows us to derive the needed length using a calculator or trig table.

Pythagorean Theorem

The Pythagorean theorem is a classic method used to solve problems involving right triangles. It's particularly useful when neither the sine, cosine, nor tangent functions directly give you the remaining unknown side. This theorem states that in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
For the longer wire:\[L_2^2 = 15^2 + 12^2\]
Here, 15 feet is the height of the pole (one leg of the triangle), and 12 feet represents the total horizontal distance from the pole to the longer wire's attachment point on the ground (second leg).

• Calculate: \(15^2 = 225\)
• Calculate: \(12^2 = 144\)
• Add these: \(225 + 144 = 369\)

The length of the wire is therefore the square root of 369:\[L_2 = \sqrt{369} \approx 19.2 \text{ feet}\]
The theorem makes it straightforward to find the hypotenuse when both legs of a triangle are known.

Angles and Distances

Angles and distances often work hand-in-hand to solve geometric problems. Understanding their relationship is crucial, especially in trigonometry. In this exercise, they define the triangle's shape and the lengths of its sides.
When you know an angle and a distance, you can use trigonometric functions to find other unknown distances. For instance, the 65-degree angle and the distance along the ground helped us determine where each wire is anchored. This ensures the wires form the correct triangle to brace the pole.

• Angle known: Use trigonometric ratios (like sine or tangent) to find unknown sides.
• Distance known: Use geometric properties like the Pythagorean theorem to find other lengths.

In conclusion, understanding the interplay between angles and distances allows for efficient problem-solving, ensuring structures like the telephone pole are securely supported.