Finding the Area

Question

A farmer is setting up separate pens for livestock by fencing off a rectangular area and running fence to create two parallel pens. If the farmer has a total of 600 feet of fence, what is the largest area that can be contained?

Area =  square feet

Answer 1

To maximize the area that the farmer can enclose with the given amount of fencing, we can use calculus to solve the problem. Let's denote:

1. x as the length of the entire rectangular area.
2. y as the width of the entire rectangular area.

The farmer is also creating two parallel pens within the rectangular area, which means that there will be two additional pieces of fencing running parallel to the width, effectively dividing the width into three sections.

Step 1: Express the total fencing used

The total amount of fencing used includes the perimeter of the rectangle plus the fencing used to create the two parallel pens. This gives us the equation:

2x+3y=600

Step 2: Express the area in terms of one variable

The area A of the rectangle is given by:

A=x×y

Using the first equation to solve for x in terms of y

x=600−3y

Substitute and find maximum of A(y).