A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 59°. A guy wire is to be attached to the top of the tower and to the ground, 88 ft downhill from the base of the tower. The angle formed by the guy wire is 22°. Find the length of the cable required for the guy wire.

To find the length of the guy wire, we can use the Law of Sines. First, let's define the given information and the unknowns.

### Given:
- The angle of inclination of the hill: \( 59^\circ \)
- The angle formed by the guy wire with the ground: \( 22^\circ \)
- The distance downhill from the base of the tower to where the guy wire will be attached: 88 feet

### Step 1: Understand the Triangle
We have a triangle where:
- One angle is \( 59^\circ \) (the hill’s inclination),
- Another angle is \( 22^\circ \) (the angle formed by the guy wire with the ground),
- The third angle is found by using the fact that the sum of the angles in a triangle is always \( 180^\circ \):
  \[
  \theta_3 = 180^\circ - 59^\circ - 22^\circ = 99^\circ
  \]
  So, the third angle is \( 99^\circ \).

### Step 2: Use the Law of Sines
The Law of Sines states:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
Where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( A \), \( B \), and \( C \) are the opposite angles.

In this case:
- \( c = 88 \) ft is the distance downhill from the base,
- \( A = 59^\circ \),
- \( B = 22^\circ \),
- \( C = 99^\circ \).

We want to find the length of the guy wire, which corresponds to the side opposite the \( 22^\circ \) angle.

\[
\frac{L}{\sin(99^\circ)} = \frac{88}{\sin(22^\circ)}
\]

### Step 3: Solve for \( L \) (the length of the guy wire)
Now, solve for \( L \):
\[
L = \frac{88 \times \sin(99^\circ)}{\sin(22^\circ)}
\]

Using a calculator:
- \( \sin(99^\circ) \approx 0.9998 \)
- \( \sin(22^\circ) \approx 0.3746 \)

\[
L = \frac{88 \times 0.9998}{0.3746} \approx \frac{87.9824}{0.3746} \approx 234.9 \, \text{ft}
\]

Thus, the length of the guy wire is approximately \( 234.9 \) feet.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the Law of Cosines, and when is it used?
2. How do you calculate the height of the tower based on this setup?
3. What other methods could you use to solve this problem?
4. How does the angle of inclination affect the length of the wire?
5. How would you approach this if the ground were flat instead of inclined?

**Tip**: When dealing with angles and triangles, double-check that your angle sum equals 180°, as it's a good way to verify your calculations.

Solve for the length of a guy wire attached to a communications tower on a 59° hill using trigonometry. This step-by-step solution applies the Law of Sines to a triangle where the angle formed by the guy wire is 22° and the distance downhill is 88 feet. Perfect for students in Grades 10-12.

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