Question

a radio tower is located 450 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 41°, and the angle of depression to the bottom of the tower is 40°. how tall is the tower? round to the nearest foot. feet question help: video message instructor submit question

Explanation:

Step1: Find 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 \(\theta = 41^{\circ}\) and adjacent side \(x = 450\) feet.
Using the formula \(\tan(41^{\circ})=\frac{h_1}{450}\), we can solve for \(h_1\):
\(h_1 = 450\times\tan(41^{\circ})\)
\(h_1\approx450\times0.8693\)
\(h_1\approx391.185\)

Step2: Find 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 \(\theta = 40^{\circ}\) and adjacent side \(x = 450\) feet.
Using the formula \(\tan(40^{\circ})=\frac{h_2}{450}\), we can solve for \(h_2\):
\(h_2 = 450\times\tan(40^{\circ})\)
\(h_2\approx450\times0.8391\)
\(h_2\approx377.595\)

Step3: Calculate the total height of the tower

The total height of the tower \(H=h_1 + h_2\)
\(H\approx391.185+377.595\)
\(H\approx768.78\)

Answer:

\(769\)