Consider the following problem: A farmer with 890 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens

Total area: 19802.5 ft2

Step-by-step explanation:

The total fencing being used will be the perimeter of the rectangular area plus 3 sides that will divide the total area in 4 pens. These sides will be parallel to the width of the total area, as the width will be the smaller side of the rectangle.

Calling the length L and the width W, we have that:

2L + 2W + 3W = 890

2L + 5W = 890

The total area will be:

A = L*W

From the first equation, we have:

L = 445 - 2.5W

Using this in the area equation, we have:

A = (445 - 2.5W)*W = 445W - 2.5W^2

To find the width that gives the maximum area, we find the vertix of the quadratic equation:

W_vertix = -b/2a

Where a and b are coefficients of the quadratic equation (in our case, a = -2.5 and b = 445). So:

W_vertix = -445/(-5) = 89

The maximum area will be:

A = (445 - 2.5W)*W = (445 - 222.5)*89 = 19802.5 ft2