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Question

The dimensions of a rectangle R are a and b, a < b. It is required to obtain a rectangle with dimensions x and y, x < a, y < a, so that its perimeter is one-third that of R, and its area is one-third that of R. The number of such (different) rectangles is:

Accepted Answer

Using the perimeter and area formulas, 2(x+y) = 23(a+b) x+y = a+b3 xy = ab3 Dividing the second equation by the last equation results in 1y + 1x = 1b + 1a Since x,y < a, 1a < 1x, 1y. Since a < b, 1b < 1a. That means 1x + 1y > 1a + 1a > 1a + 1b This is a contradiction, so there are 0 rectangles that satisfy the conditions.