Maximizing Rectangular Park Area

CAT 2017 Slot 2 · QA · Hard · Geometry

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If three sides of a rectangular park have a total length 400 ft., then the area of the park is maximum when the length (in ft.) of its longer side is

Answer

200

Explanation

Let the dimensions of the rectangle be xx and yy.

Three sides total 400 ft. Thus, 2x+y=400  ⟹  y=400−2x2x + y = 400 \implies y = 400 - 2x.

Area A=xy=x(400−2x)=400x−2x2A = x y = x(400 - 2x) = 400x - 2x^2.

This quadratic expression reaches maximum at x=−4002(−2)=100x = \frac{-400}{2(-2)} = 100.

Then y=400−2(100)=200 fty = 400 - 2(100) = 200\text{ ft}.

The longer side is y=200y = 200 ft.

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