A farmer has 180 feet of fence and wants a pen to adjoin to the whole side of the 120 foot barn as shown.

What should the dimentions be for maximum area. Note that in this case

Write the equation for perimeter of the three sided rectangle. Write the equation for area in terms of l, w.

You should have enough to solve. I will be happy to critique your work.

To determine the dimensions of the pen that will result in the maximum area, we need to find the length (l) and width (w) that will optimize the given constraints.

Let's start by setting up the equation for the perimeter of the three-sided rectangle. The perimeter is the sum of the lengths of all three sides.

In this case, the rectangle comprises two sides of length w and one side of length (120 - w). Therefore, the perimeter equation is:

Perimeter = 2w + (120 - w)

Simplifying the equation, we get:

Perimeter = 2w + 120 - w

Next, we have the information that the total amount of fence available is 180 feet. Since the perimeter of the pen corresponds to the amount of fence used, we can set up the equation:

2w + 120 - w = 180

By solving this equation, we can find the value of w (width) that maximizes the pen's area.

Once we have the width, we can find the length by subtracting the width from the barn's length:

Length = 120 - w

Now, let's move on to the equation for the area of the pen.

The area of the rectangle is given by the product of its length and width:

Area = length × width

Substituting the values we found earlier, the equation becomes:

Area = (120 - w) × w

We can simplify this equation further if needed, but for now, we have the required equations to solve for the dimensions that will result in the maximum area.

To find the values of w and the corresponding length, we need to solve the equation 2w + 120 - w = 180.