Question

a radio tower is located 325 feet from a building. from a window in the building, a person determines that the angle of elevation to the top of the tower is 37° and that the angle of depression to the bottom of the tower is 29°. how tall is the tower? feet add work

Explanation:

Step1: Calculate the height from the window to the top of the tower

Let the height from the window to the top of the tower be \(h_1\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Given \(\theta = 37^{\circ}\) and adjacent side \(x = 325\) feet.
Using the formula \(\tan37^{\circ}=\frac{h_1}{325}\), so \(h_1 = 325\times\tan37^{\circ}\).
Since \(\tan37^{\circ}\approx0.7536\), then \(h_1=325\times0.7536 = 244.92\) feet.

Step2: Calculate the height from the window to the bottom of the tower

Let the height from the window to the bottom of the tower be \(h_2\).
Given \(\theta = 29^{\circ}\) and adjacent side \(x = 325\) feet.
Using the formula \(\tan29^{\circ}=\frac{h_2}{325}\), so \(h_2 = 325\times\tan29^{\circ}\).
Since \(\tan29^{\circ}\approx0.5543\), then \(h_2=325\times0.5543=180.1475\) feet.

Step3: Calculate the total height of the tower

The total height of the tower \(H=h_1 + h_2\).
\(H=244.92+180.1475 = 425.0675\approx425.07\) feet.

Answer:

\(425.07\)