Question

A rancher wants to construct a rectangular pen by extending an existing 100 ft stone wall by an additional x feet of fencing. If the farmer has a total of 700 ft of fencing to complete the pen, find the dimensions that will maximize the enclosed area.

Answer 1

175ft by 175ft

Explanation:

Let the length of the rectangular pen be L

let the width be W

Perimeter of the pen P = 2L+2W

Area A = LW

Given

P = 700ft

L = 100+x

P = 2(100+x) + 2w

P = 200+2x+2w

700 = 200+2x+2w

500 = 2x+2w

250 = x+ w

w = 250-x..... 1

A = (100+x)w ....2

substitute 1 into 2;

A = (100+x)(250-x)

A = 25000-100x+250x-x²

A = 25000+150x-x²

To maximize the area, dA/dx = 0

dA/dx = 150-2x

150-2x = 0

150 = 2x

x = 150/2

x = 75ft

Since

x+w = 250

75+w = 250

w = 250-75

w = 175ft

The width of the rectangular pen will be 175ft

The length will be 100+x = 100+75 = 175ft

Hence the dimension that will maximize the enclosed area is 175ft by 175ft