2008 AMC 10A Problem 14

Problem 14 of 25IntermediateAlgebra Geometry

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 screen is

4:3

with a

27

-inch diagonal,

h : w : 27 = 3 : 4 : 5,

giving height

h = \dfrac{3}{5}\cdot 27 = 16.2

and width

w = \dfrac{4}{5}\cdot 27 = 21.6.

The lit

2:1

region has the full width

21.6

and height

\dfrac{21.6}{2} = 10.8.

The two strips share the remaining height, so each has height

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

Thus, the correct answer is D.

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