Problem:
Older television screens have an aspect ratio of 4:3. That is, the ratio of the width to the height is . The aspect ratio of many movies is not , 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let and be the height and width of the screen, respectively, in inches. By the Pythagorean Theorem, , so
The height of the non-darkened portion of the screen is half the width, or 10.8 inches. Therefore the height of each darkened strip is
OR
The screen has dimensions for some . The portion of the screen not covered by the darkened strips has aspect ratio , so it has dimensions . Thus the darkened strips each have height . By the Pythagorean Theorem, the diagonal of the screen is inches. Hence the height of each darkened strip is inches.
The problems on this page are the property of the MAA's American Mathematics Competitions