Question

a wire connects the top of a 128 foot tall cell tower to the ground. if the angle between the wire and the ground is 76°, how long is the wire? feet round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the height of the tower (opposite side) is \(h = 128\) feet and the wire is the hypotenuse (\(l\)). The angle between the wire and the ground is \(\theta=76^{\circ}\). Using the sine function: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin(76^{\circ})=\frac{128}{l}\).

Step2: Solve for \(l\)

Rearrange the formula to get \(l=\frac{128}{\sin(76^{\circ})}\).
We know that \(\sin(76^{\circ})\approx0.9703\). Then \(l=\frac{128}{0.9703}\).
Calculate \(\frac{128}{0.9703}\approx131.92\).

Answer:

\(131.92\)