Questions: A guy 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 degrees, how long is the wire? [?] feet Round your answer to the nearest hundredth.

A guy 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 degrees, how long is the wire? [?] feet Round your answer to the nearest hundredth.
Transcript text: A guy wire connects the top of a 128 foot tall cell tower to the ground. If the angle between the wire and the ground is $76^{\circ}$, how long is the wire? [?] feet Round your answer to the nearest hundredth.
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Solution

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Solution Steps

Step 1: Calculate \( \sin(76^{\circ}) \)

To find the length of the guy wire, we first need to calculate \( \sin(76^{\circ}) \). The value obtained is: \[ \sin(76^{\circ}) \approx 0.9703 \]

Step 2: Use the Sine Function to Find the Length of the Wire

Using the sine function, we can relate the height of the tower to the length of the wire. The formula is given by: \[ \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} \] Substituting the known values: \[ \text{hypotenuse} = \frac{128}{\sin(76^{\circ})} \approx \frac{128}{0.9703} \approx 131.92 \]

Final Answer

The length of the wire is approximately: \[ \boxed{131.92 \text{ feet}} \]

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