Draw a picture and label it. Set up an equation and solve it. Round the measure of each angle to the nearest tenth of a degree. Round the length of a side to the nearest tenth of a unit. Show all your work. Assume the ground is level unless indicated otherwise.
A guy wire whose grounded end is 16 ft from the telephone pole it supports makes an angle of 54.6° with the ground.


Here is the diagram of the right triangle formed by the telephone pole, the guy wire, and the ground. Now, let's set up the equation and solve for the height of the pole and the length of the guy wire.

### Step 1: Identify the triangle elements
- Angle with the ground: \( \theta = 54.6^\circ \)
- Distance from the base of the pole to the guy wire’s grounded end: adjacent side = 16 feet
- Height of the pole = opposite side
- Length of the guy wire = hypotenuse

### Step 2: Use trigonometric ratios
To find the height of the pole, we use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{16}
\]
Solving for \( h \):
\[
h = 16 \times \tan(54.6^\circ)
\]
Using a calculator:
\[
h \approx 16 \times 1.406 \approx 22.5 \text{ feet}
\]
So, the height of the pole is approximately 22.5 feet.

### Step 3: Find the length of the guy wire (hypotenuse)
Use the cosine function:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{16}{L}
\]
Solving for \( L \):
\[
L = \frac{16}{\cos(54.6^\circ)}
\]
Using a calculator:
\[
L \approx \frac{16}{0.581} \approx 27.5 \text{ feet}
\]
So, the length of the guy wire is approximately 27.5 feet.

### Final Answers:
- Height of the pole: \( 22.5 \) feet
- Length of the guy wire: \( 27.5 \) feet

Would you like any additional details or have questions?

Here are 5 related questions to deepen your understanding:
1. How would the height change if the angle was increased to 60°?
2. What would happen to the length of the guy wire if the distance from the pole was increased?
3. How can you use the sine function to verify the length of the guy wire?
4. How does changing the angle of the wire affect the stability of the structure?
5. How could you apply this problem to a real-world scenario involving a different structure?

**Tip:** Understanding the relationships between angles and sides in right triangles is crucial for solving various geometry and trigonometry problems.

Draw a picture and label it. Set up an equation and solve it. Round the measure of each angle to the nearest tenth of a degree. Round the length of a side to the nearest tenth of a unit. Show all your work. Assume the ground is level unless indicated otherwise. A guy wire whose grounded end is 16 ft from the telephone pole it supports makes an angle of 54.6° with the ground.

Learn how to solve for the height of a telephone pole and the length of a guy wire using trigonometric ratios. In this problem, the angle of 54.6° between the guy wire and the ground, along with the 16 ft distance from the pole, provides key data. Suitable for students in Grades 9-11.

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