Please answer my question with steps.

Question

A pig farmer has 1200 ft of fencing and wants to fence off a rectangular field. What are the dimensions of the field that has the largest area? (Use calculus to answer the question.)

Answer 1

Let L = length of rectangle

Let W = width of rectangle 

 

Perimeter = 1200 feet = 2*L + 2*W  ==>  L = 600 - W

 

Area = L*W = (600 - W) * W = -W^2 + 600W  

 

To maximize the area take the derivatives of the area function with respect to W  

 

A' = -2*W + 600

A'' = -2 which is negative so any critical point will be a maximum

 

Set A' = 0  ==>  -2*W + 600 = 0

-2*W = -600

W = 300

 

So L = 600 - 300 = 300

 

The field with the largest area has dimensions 300 ft by 300 ft