Question

a radio tower is located 400 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 34° and that the angle of depression to the bottom of the tower is 25°. how tall is the tower? round answer to one decimal place.

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 the angle of elevation \(\theta = 34^{\circ}\) and the adjacent side (distance from building to tower) \(x = 400\) feet. Using the formula \(h_1=x\tan\theta\), we have \(h_1 = 400\times\tan(34^{\circ})\). Since \(\tan(34^{\circ})\approx0.6745\), then \(h_1=400\times0.6745 = 269.8\) 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 the angle of depression \(\alpha=25^{\circ}\). Using the formula \(h_2=x\tan\alpha\) (because the angle of depression is equal to the angle of elevation from the bottom of the tower to the window). Since \(\tan(25^{\circ})\approx0.4663\) and \(x = 400\) feet, then \(h_2=400\times0.4663=186.52\) feet.

Step3: Calculate the total height of the tower

The total height of the tower \(H=h_1 + h_2\). Substitute \(h_1 = 269.8\) and \(h_2=186.52\) into the formula. \(H=269.8+186.52=456.32\approx456.3\) feet.

Answer:

\(456.3\) feet