Question

a guy wire is secured near the top of a television transmitting tower. the guy wire meets the ground at an angle of 52°. if the height of the tower is 27 ft, how long is the guy wire? round to two decimal places as needed.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the height of the tower is the opposite side to the angle of \(52^\circ\), and the guy wire is the hypotenuse. We use the sine function, which is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). Let \(h\) be the height of the tower (\(27\) ft), \(L\) be the length of the guy wire (hypotenuse), and \(\theta = 52^\circ\). So, \(\sin(52^\circ)=\frac{h}{L}\).

Step2: Solve for \(L\)

Rearranging the formula for \(L\), we get \(L=\frac{h}{\sin(52^\circ)}\). We know \(h = 27\) ft, and \(\sin(52^\circ)\approx0.7880\) (using a calculator). Plugging in the values, we have \(L=\frac{27}{0.7880}\).

Step3: Calculate the length

\(L=\frac{27}{0.7880}\approx34.26\) (rounded to two decimal places).

Answer:

The length of the guy wire is approximately \(\boldsymbol{34.26}\) feet.