Bunuel wrote:
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?
(A) 2
(B) 2.25
(C) 2.5
(D) 2.7
(E) 3
First, let’s find the width and height of the TV screen. Since the TV screen has an aspect ratio of 4:3, let the width be 4x and the height be 3x, for some x. Since we know that the diagonal is 27, we can use the Pythagorean theorem to set up the equation:
(4x)^2 + (3x)^2 = 27^2
16x^2 + 9x^2 = 27^2
25x^2 = 27^2
x^2 = 27^2/25
x = √(27^2/25) = 27/5
So the width of the screen is 4(27/5) = 108/5, and the height is 3(27/5) = 81/5. Now, since the movie has an aspect ratio of 2:1, its width will match that of the screen. That is, the width of movie is 108/5, but its height will be only half of this, or 54/5. Since the height of the screen is 81/5 and the height of the movie is 54/5, the height of each of the two darkening strips is (81/5 - 54/5)/2 = (27/5)/2 = 27/10 = 2.7.
Answer: D
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