Question

abdul is sitting in a movie theater, 13 meters from the screen. the angle of elevation from his line of sight to the top of the screen is 10°, and the angle of depression from his line of sight to the bottom of the screen is 53°. find the height of the entire screen. do not round any intermediate computations. round your answer to the nearest tenth. note that the figure below is not drawn to scale.

Explanation:

Step1: Find height below line of sight

We know the horizontal distance from Abdul to the screen is 13 meters (adjacent side for the angle of depression). Using the angle of depression \(53^\circ\), the height below his line of sight (\(h_1\)) can be found using \(\tan(53^\circ)=\frac{h_1}{13}\). So \(h_1 = 13\times\tan(53^\circ)\). Calculating \(\tan(53^\circ)\approx1.3333\), so \(h_1\approx13\times1.3333 = 17.3329\) meters.

Step2: Find height above line of sight

Using the angle of elevation \(10^\circ\), the height above his line of sight (\(h_2\)) is found by \(\tan(10^\circ)=\frac{h_2}{13}\). So \(h_2 = 13\times\tan(10^\circ)\). Calculating \(\tan(10^\circ)\approx0.1763\), so \(h_2\approx13\times0.1763 = 2.2919\) meters.

Step3: Total height of screen

The total height (\(H\)) is \(h_1 + h_2\). So \(H\approx17.3329+2.2919 = 19.6248\) meters. Rounding to the nearest tenth gives \(19.6\) meters.

Answer:

\(19.6\) meters