A farmer wants to fence an area of 1350 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 amount of fencing needed?

The lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 45 feet by 30 feet

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

Since the area is 1350, hence:

Area = length * width = x * y

1350 = xy

y = 1350/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(1350/x) = 2x + 4050/x

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

dP/dx = 2 - 4050/x²

2 - 4050/x² = 0

2 = 4050/x²

2x² = 4050

x² = 2025

x = 45 feet

y = 1350/x = 1350/45 = 30 feet

Hence, the lengths of the sides of the rectangular field so as to minimize the amount of fencing needed is 45 feet by 30 feet

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Length of rectangular field=45 ft

Width of rectangular field=30 ft

Step-by-step explanation:

We are given that

Area of rectangular field,A=1350 square feet

Let

Length of rectangular field=x

Width of rectangular field=y

a fence out parallel to y

Fence used,P=2x+2y+y=2x+3y

Area of rectangular field,

Differentiate w.r.t x

It is positive because length cannot be negative.

Again, differentiate w.r.t x

Fencing needed is minimum at x=45

Substitute the value

Hence, Length of rectangular field=45 ft

Width of rectangular field=30 ft

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