2008 AMC 12A Problem 9

Older television screens have an aspect ratio of

4:3.

That is, the ratio of the width to the height is

4:3.

The aspect ratio of many movies is not

4:3,

so they are sometimes shown on a television screen by “letterboxing” — darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of

2:1

and is shown on an older television screen with a

27

-inch diagonal. What is the height, in inches, of each darkened strip?

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Solution

Since the sides and diagonal are in ratio

3:4:5,

the height is

\tfrac{3}{5} \cdot 27 = 16.2

inches and the width is

\tfrac{4}{5} \cdot 27 = 21.6

inches. The movie has aspect ratio

2:1,

so its height is

\tfrac{21.6}{2} = 10.8

inches. Each darkened strip therefore has height

\dfrac{16.2 - 10.8}{2} = 2.7

inches. Thus, D is the correct answer.

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