CAT 2021 Slot 3 · Quant · Question 19

CAT 2021 Slot 3 previous year question on Maxima and Minima, with the answer and a step-by-step solution.

Maxima and MinimaAlgebraMultiple choiceMedium
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 ₹, is
  1. A
    90000
  2. B
    160000
  3. C
    120000
  4. D
    100000
Pick an option, then check.
Answer & solution

Answer: Option C

Let the expensive side have length L and the other dimension be W, so LW=60000.

The side opposite the expensive one also has length L but costs 100 per ft, and the two sides of length W cost 100 per ft each. So the total cost is

C=200L+100L+2\left( 100W \right)=300L+200W.

Substituting W=\frac{60000}{L} gives C=300L+\frac{12000000}{L}.

Both terms are positive, so by AM-GM their sum is least when they are equal, and C\ge 2\sqrt{300L\cdot \frac{12000000}{L}}=2\sqrt{3600000000}=2\times 60000=120000.

The minimum ₹120000 is reached when 300L=\frac{12000000}{L}, that is L=200 ft and W=300 ft.

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