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 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.

2008 AMC 10A Problem 14

Solution:

Problem 14 in Other Years