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 32°, and that the angle of depression to the bottom of the tower is 21°. how tall is the tower? (round your answer to four decimal places.) ft resources ebook tutorial determine length of right triangle side graphing the sine function details my notes ask your teacher practice another oscat1 7.2.120.tut

Explanation:

Step1: Analyze the right - triangle situation

We have a right - triangle problem. The horizontal distance from the building to the tower is \(d = 400\) feet. Let the height from the level of the window to the top of the tower be \(h_1\) and the height from the level of the window to the bottom of the tower be \(h_2\).

Step2: Use the tangent function for the top - part

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For the angle of elevation \(\theta_1 = 32^{\circ}\), \(\tan\theta_1=\frac{h_1}{400}\), so \(h_1 = 400\times\tan(32^{\circ})\).

Step3: Calculate \(h_1\)

\(h_1=400\times\tan(32^{\circ})\approx400\times0.6249 = 249.96\) feet.

Step4: Use the tangent function for the bottom - part

For the angle of depression \(\theta_2 = 21^{\circ}\), \(\tan\theta_2=\frac{h_2}{400}\), so \(h_2 = 400\times\tan(21^{\circ})\).

Step5: Calculate \(h_2\)

\(h_2=400\times\tan(21^{\circ})\approx400\times0.3839 = 153.56\) feet.

Step6: Find the height of the tower \(H\)

The height of the tower \(H=h_1 + h_2\). \(H=249.96+153.56=403.52\) feet.

Answer:

\(403.5200\)