<p>Older television screens have an aspect ratio of <img alt="$4: 3$" height="14" src="https://latex.artofproblemsolving.com/a/9/a/a9a03b059b84a40291c4c20ea6a12c7127bff4c3.png" width="35" />. That is, the ratio of the width to the height is <img alt="$4: 3$" height="14" src="https://latex.artofproblemsolving.com/a/9/a/a9a03b059b84a40291c4c20ea6a12c7127bff4c3.png" width="35" />. The aspect ratio of many movies is not <img alt="$4: 3$" height="14" src="https://latex.artofproblemsolving.com/a/9/a/a9a03b059b84a40291c4c20ea6a12c7127bff4c3.png" width="35" />, 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 <img alt="$2: 1$" height="12" src="https://latex.artofproblemsolving.com/c/3/8/c38896966af12256c9dae24b447911c290015bfc.png" width="32" /> and is shown on an older television screen with a <img alt="$27$" height="14" src="https://latex.artofproblemsolving.com/1/0/5/1051e7d41ec0b8bd688a872be17538413f4ee97b.png" width="20" />-inch diagonal. What is the height, in inches, of each darkened strip?</p> <p><img alt="[asy] unitsize(1mm); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,0)--(21.6,0)--(21.6,16.2)--(0,16.2)--cycle); [/asy]" height="78" src="https://latex.artofproblemsolving.com/0/e/0/0e0eb295ca042287f3d625b9ac6eed49532ef1ab.png" width="105" /></p>

Question No: 274

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?

$[asy] unitsize(1mm); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,0)--(21.6,0)--(21.6,16.2)--(0,16.2)--cycle); [/asy]$

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