2006 iTest Problems/Problem 26
Problem
A rectangle has area and perimeter . The largest possible value of can be expressed as , where and are relatively prime positive integers. Compute .
Solution
Let and be the length and width of the rectangle, respectively. The area of the rectangle is , and the perimeter of the rectangle is . We wish to maximize .
By the AM-GM Inequality, , with equality occurring when . Multiply both sides by 4 to get .
Because the length and width of a rectangle is positive, . Thus, squaring both sides would not affect the inequality sign, so . Finally, since is positive, we can divide both sides by and to get . The equality case satisfies the equality statement, so the highest possible ratio is . Therefore, .
See Also
2006 iTest (Problems, Answer Key) | ||
Preceded by: Problem 25 |
Followed by: Problem 27 | |
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