Question

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.

Answer 1

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(theta) = 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(68.96)=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.