Lowest cost of fencing rectangular plot

CAT 2021 Slot 3 · QA · Easy · Mensuration

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The cost of fencing a rectangular plot is ₹ 200 per ft along one side, and ₹ 100 per ft along the three other sides. If the area of the rectangular plot is 60000 sq. ft, then the lowest possible cost of fencing all four sides, in INR, is

  1. A.

    90000

  2. B.

    160000

  3. C.

    120000

  4. D.

    100000

Answer

C

Explanation

Let dimensions of rectangle be xx and yy with area xy=60000xy = 60000. Let side xx have fencing rate of ₹ 200 per ft. The opposite side xx costs ₹ 100 per ft, and remaining sides y,yy, y cost ₹ 100 per ft each.

Total cost C=200x+100x+100y+100y=300x+200yC = 200x + 100x + 100y + 100y = 300x + 200y.

By AM-GM Inequality: 300x+200y≥2(300x)(200y)=260000xy=260000×60000=120000300x + 200y \ge 2\sqrt{(300x)(200y)} = 2\sqrt{60000 xy} = 2\sqrt{60000 \times 60000} = 120000

Lowest possible cost is ₹ 120000.

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