Question

The back of Alisha's property is a creek. Alisha would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture. If there is 600 feet of fencing available, what is the maximum possible area of the pasture?

Answer 1

A square has all 4 sides equal, so divide the amount of fence available by 4 to get the length of one side of the square

600/4 = 150

Now since the creek is being used for one side, add one side of the square to the other side to get a rectagle 150 by 300 feet.

Area = 150 x 300 = 45,000 square feet.

Answer 2

The maximum possible area of the pasture is;

A_max = 45000 ft²

• We are given;

Available fencing; Perimeter = 600 feet

Number of sides to fence; 3 sides of rectangle

 

• Since we are dealing with rectangle, let L be the length and W be the width.

Perimeter of rectangle; P = 2L + 2W

But we are told one of the edges is the creek.

Thus, New perimeter = L + 2W

thus, we have;  L + 2W = 600

L = 600 - 2W

     

• Formula for Area of a rectangle is; A = LW

Let's put 600 - 2W for L in the area equation to get;

A = (600 - 2W)W

A = 600W - 2W²

• We can maximize this area by finding the value of W when dA/dW = 0

Thus;

dA/dW = 600 - 4W

At dA/dW = 0, we have;

600 - 4W = 0

4W = 600

W = 600/4

W = 150 ft

Let's put 150 for W in L = 600 - 2W

L = 600 - 2(150)

L = 600 - 300

L = 300 ft

 Therefore, Maximum possible area of pasture = 300 × 150 = 45000 ft²

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