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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10, SAT-II.
If the area of a rectangle varies linearly with the area of the circle that is inscribed in it, then the length of the base of the rectangle varies with the length of its width:
The circle that is inscribed in the rectangle touches at most three sides, unless the rectangle is a square. The diameter of the circle must be equal to the smaller side of the rectangle. Let us assume, without loss of generality, that it is the width.
The area of the circle is:
and the problem states that the area of the rectangle is directly proportional to the area of the circle:
and re-arrange to show that:
the base and the width vary linearly (are directly proportional) in this case.