A farmer builds a rectangular grid of pens with 1 row and 5 columns using 650 feet of fencing. What dimensions will maximize the total area of the pen?

Question

What dimensions will maximize the total area of the pen? Width of each row of the pens ??? Height of each column of the pen??

Answer 1

First, it's good to draw this out to visualize the grid and help us set up our equations. This is a terrible diagram, but please note I'm using r for the height of the row and c for the width of the column, but this is arbitrary.

____________

r |__|__|__|__|__|

c

Now, we can translate the information presented in the problem into equations. First, an equation for the perimeter, both outside and inside. If you count the line segments in the diagram, you see that we'll need 6 lengths of r between pens and 10 lengths of c on the top and bottom. We know the total perimeter has to be 650 feet. So, our equation is 6r+10c=650, with r and c in feet.

Next, we can set up an expression for total area. This is simpler, just a rectangle with dimensions r by 5c, so our equation is r x 5c = A, since we don't know what area we need yet. Well, that equation has three variables, so we can't find a maximum yet. We'll have to use the perimeter equation to find either r or c in terms of the other and plug that in.

I'm going to find c in terms of r, but you could do it either way. When I solve 6r+10c=650 for c, I get c=65-0.6r. Plugging that into the area equation gives r x 5(65-0.6r) = A, or A = -3r² + 325r.

Now, you might recognize this as a polynomial function that will have a maximum, and we can finally use calculus. To find the maximum, we set the derivative of that bolded equation equal to 0, so -6r + 325 = 0. Solved for r, now we know r = 54 1/6 feet. We can figure out c from that, using c = 65-0.6(54 1/6) = 32.5.

Height of a row: 54 feet, 2 inches

Width of a column: 32 feet, 6 inches (total width is 5 columns, or 162 feet, 6 inches)

Answer 2

162.5/12 = 54 1/6 feet wide

h= 650/20 = 32 1/2 feet high