# How do you approximate the height of the screen to the nearest tenth?

## You want to determine the height of the screen at a drive-in movie theater. You use a cardboard square to line up the top and bottom of the screen structure. The vertical distance from the ground to your eye is 5 feet and the horizontal distance from you to the screen is 13 feet. The bottom of the screen is 6 feet from the ground.

May 29, 2018

32.8 feet

#### Explanation:

Since the bottom triangle is right-angled, Pythagoras applies and we can calculate the hypotenuse to be 12 (by $\sqrt{{13}^{2} - {5}^{2}}$ or by the 5,12,13 triplet).

Now, let $\theta$ be the smallest angle of the bottom mini triangle, such that

$\tan \left(\theta\right) = \frac{5}{13}$ and thus $\theta = {21.03}^{o}$

Since the big triangle is also right-angled, we can thus determine that the angle between the 13 foot side and the line connecting to the top of the screen is $90 - 21.03 = {68.96}^{o}$.

Finally, setting $x$ to be the length from the top of the screen to the 13 foot line, some trigonometry gives

$\tan \left(68.96\right) = \frac{x}{13}$ and therefore $x = 33.8$ feet.

Since the screen is 1 foot above the ground, and our calculated length is from the person's eye height to the top of the screen, we must subtract 1 foot from our $x$ to give the height of the screen, which is $32.8$ feet.