Question

salma is sitting in a movie theater, 8 meters from the screen. the angle of elevation from her line of sight to the top of the screen is 11°, and the angle of depression from her line of sight to the bottom of the screen is 51°. 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 the height from the bottom of the screen to her eye level

Use the tangent function: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
For the $51^{\circ}$ angle, let the height from the bottom of the screen to her eye level be $h_1$.
We have $\tan(51^{\circ})=\frac{h_1}{8}$.
So, $h_1 = 8\times\tan(51^{\circ})$.
Using a calculator, $\tan(51^{\circ})\approx1.2349$, then $h_1\approx8\times1.2349 = 9.8792$.

Step2: Find the height from the top of the screen to her eye level

For the $11^{\circ}$ angle, let the height from the top of the screen to her eye level be $h_2$.
Using the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan(11^{\circ})=\frac{h_2}{8}$.
Since $\tan(11^{\circ})\approx0.1944$, then $h_2 = 8\times0.1944=1.5552$.

Step3: Calculate the height of the screen

The height of the screen $H=h_1 - h_2$.
Substitute the values of $h_1$ and $h_2$: $H=9.8792-1.5552 = 8.324$.

Answer:

$8.3$ meters