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Maximizing Rectangular Park Area

CAT 2017 Slot 2 · Quantitative Ability · Hard · Geometry

This is a hard Quantitative Ability question from the CAT 2017 Slot 2 paper. It tests Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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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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