Question

A radio tower has a cable that anchors it to the ground. The cable is 300 yd. long and meets the ground at an angle of elevation of 64°
. Find the distance from the cable to the base of the tower (x) and the height of the tower (y), rounding your answer to 2 decimal places. You must show all work in order to receive credit. You may type your work or upload a picture. (4 points)

Answer 1

To solve this problem, we can use some trigonometry. The radio tower, the cable, and the ground form a right triangle, with the tower as one side (height, \(y\)), the ground directly beneath the cable as the other side (distance from cable to the base of the tower, \(x\)), and the cable as the hypotenuse.

Given:

- Length of the cable (hypotenuse) = 300 yds
- Angle of elevation = 64°

We can use the following trigonometric relationships:

1. **Sine Function**:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \implies \sin(64^\circ) = \frac{y}{300}
\]

2. **Cosine Function**:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \implies \cos(64^\circ) = \frac{x}{300}
\]

### Step 1: Find the height of the tower (\(y\))

Using the sine function:
\[
y = 300 \cdot \sin(64^\circ)
\]

Calculating \(y\):
- First, find \(\sin(64^\circ)\):
\[
\sin(64^\circ) \approx 0.8988 \quad (\text{using a calculator})
\]

Then:
\[
y \approx 300 \cdot 0.8988 \approx 269.64 \text{ yds}
\]

### Step 2: Find the distance from the cable to the base of the tower (\(x\))

Using the cosine function:
\[
x = 300 \cdot \cos(64^\circ)
\]

Calculating \(x\):
- First, find \(\cos(64^\circ)\):
\[
\cos(64^\circ) \approx 0.4384 \quad (\text{using a calculator})
\]

Then:
\[
x \approx 300 \cdot 0.4384 \approx 131.52 \text{ yds}
\]

### Final Results

- Distance from the cable to the base of the tower (\(x\)): **131.52 yds**
- Height of the tower (\(y\)): **269.64 yds**

Thus, rounding to two decimal places:

\[
\boxed{x \approx 131.52 \text{ yds}, \quad y \approx 269.64 \text{ yds}}
\]