Question

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?

$2$

$2.25$

$2.5$

$2.7$

$3$

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.

2008 AMC 12A Problem 9

Solution:

Problem 9 in Other Years