Question

A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. what should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?

Answer 1

3x +2y A =xy = 6,000,000 y = 6,000, 000/x L = 2x + 12,000,000/x 3- (12,000,000/x^2) = 0 3 = 12, 000,000/x^2 sqrt x^2 = sqrt 4,000,000 x = 2,000 and y = 3,000 

Answer 2

The lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 3000 feet by 6000 feet

Let x represent the length of the fence and y represent the width of the fence.

Since the area is 6000000, hence:

Area = length * width = x * y

6000000 = xy

y = 6000000/x

A fence divide the field in half and is parallel to one of the sides of the rectangle. Hence:

Amount of fencing needed (P) = x + x + y + y  + y = 2x + 3y

Amount of fencing needed (P) = 2x + 3(6000000/x) = 2x + 18000000/x

To minimize the amount of fencing needed, dP/dx = 0, hence:

dP/dx = 2 - 18000000/x²

2 - 18000000/x² = 0

2 = 18000000/x²

2x² = 18000000

x² = 9000000

x = 3000 feet

y = 18000000/x = 18000000/3000 = 6000 feet

Hence, the lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 3000 feet by 6000 feet

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