Question

a radio transmission tower is 200 feet high. how long should a guy wire be if it is to be attached to the tower 13 feet from the top and is to make an angle of 45° with the ground?
the guy wire should be \\( \boxed{} \\) feet long.
(do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Find the height of the attachment point

The tower is 200 feet high, and the guy wire is attached 13 feet from the top. So the height from the ground to the attachment point is $200 - 13 = 187$ feet.

Step2: Use trigonometry to find the length of the guy wire

We have a right triangle where the height (opposite side to the $45^{\circ}$ angle) is 187 feet, and the guy wire is the hypotenuse. We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\theta = 45^{\circ}$, $\sin(45^{\circ})=\frac{\sqrt{2}}{2}$. Let $L$ be the length of the guy wire. Then $\sin(45^{\circ})=\frac{187}{L}$. Solving for $L$, we get $L=\frac{187}{\sin(45^{\circ})}$. Since $\sin(45^{\circ})=\frac{\sqrt{2}}{2}$, this becomes $L = \frac{187}{\frac{\sqrt{2}}{2}}=187\times\frac{2}{\sqrt{2}} = 187\sqrt{2}$. Calculating the numerical value, $187\sqrt{2}\approx187\times1.41421356\approx264.46$.

Answer:

264.46