Multiple choice

A school plans to develop a rectangular botanical garden on its campus. To protect the garden, the authorities decide to fence it on all four sides. However, the side facing the main road is to be fenced with a premium decorative iron fence that costs Rs. 300 per foot, while the other three sides will use standard fencing costing Rs. 150 per foot. The total area of the garden is 1,35,000 square feet, and the school wants to minimise the total fencing cost while enclosing this area. What is the minimum total fencing cost (in INR) of all the four sides of the garden?

  1. 3,66,000

  2. 2,27,000

  3. 2,70,000

  4. 3,60,000
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let length be L and width be W. Area LW = 135000. Cost = 300L + 150L + 150W + 150W = 450L + 300W. To minimize, use AM-GM: 450L + 300W >= 2 * sqrt(450L * 300W) = 2 * sqrt(135000 * 135000) = 2 * 135000 = 270000.

AI explanation

Let the side with the premium fence have length y and the other dimension be x, so the area equation is xy = 135,000 and the total cost is 300y + 150y + 150x + 150x, which simplifies to C = 450y + 300x. Substituting x = 135,000 divided by y into the cost equation gives C = 450y + 40,500,000 divided by y, and setting its derivative to zero yields 450 = 40,500,000 divided by y squared, making y equal to 300. Using this dimension, the cost is 450 times 300 plus 300 times 450, resulting in a minimum total cost of 2,70,000.

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