Question

A farmer wants to fence an area of 13.5 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

  13.5 = lw 

13.5/l = w 

3l + 2w = C 

C = 3l + 2*(13.5/l) 

C = 3l + (27/l) 

dC(l)/dl = 0 

3 - (27/l^2) = 0 

3*(l^2) - 27 = 0 

(l^2) - 9 = 0 

(l - 3)*(l + 3) = 0 

l = 3 

13.5 = 3w 

l = 3000; w = 4500

Therefore, to minimize the cost of the fence, length should be 3000 ft while width should be 4500 ft.

Answer 2

The lengths of the sides of the rectangular field that will minimize the cost of the fence is;

3674.26 ft

Let the dimensions be;

Length = L

Width = w

Now, formula for perimeter of a rectangle is;

P = 2L + 2w

Formula for area of a rectangle is;

A = Lw

Thus;

w = A/L

Put A/L for W in perimeter equation to get;

P = 2L + 2(A/L)

P = 2L + 2A/L

Differentiating with respect to L gives;

dP/dL = 2 - 2A/L²

At dP/dL = 0;

2 - 2A/L² = 0

2A/L² = 2

L = √A

Put √A for L in the area equation to get;

A = w√A

w = A/√A

Rationalizing the denominator according to surds gives us;

w = √A

This means L = w = √A

Since length is equal to width, then it is a square.

We are told that the area is 13.5 million ft²

Thus;

L = w = √(13.5 × 10^(6))

L = w = 3674.26 ft

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