A cattle rancher wants to enclose a rectangular area and then divide it into six pens with fencing parallel to one side of the rectangle (see the figure below). There are 540 feet of fencing available to complete the job. What is the largest possible total area of the six pens?
as always, maximum area is when the fence is divided equally among lengths and widths. So, since I cannot see the diagram, but it implies the pens are in a row, we have six pens of length x and width y.
2x+7y = 540
area = 6xy = 6(540-7y)/2 * y
= 1620y - 21y^2
This is a parabola with vertex (maximum area) at y=270/7
So, each pen is 135 by 270/7
Note how the fence is equally divided among lengths and widths.
To find the largest possible total area of the six pens, we need to determine the dimensions of the rectangle that will maximize the area.
Let's assume the length of the rectangular enclosure is L and the width is W. Since there are six pens, we can divide the length into three equal parts and the width into two equal parts.
So, each pen's length would be L/3, and its width would be W/2.
Now, let's calculate the total length of the fencing needed for the six pens:
For the length, we need two sides of each pen plus the side connecting the pens. Since there are three pens in the length, the total length of fencing for the length is:
2 * (L/3) * 3 = 2L
For the width, we only need one side of each pen. Since there are two pens in the width, the total length of fencing for the width is:
2 * (W/2) * 2 = 2W
The total length of fencing needed is the sum of the lengths from the length and the width:
2L + 2W = 540
Now, we can rearrange the equation to solve for either L or W in terms of the other variable. Let's solve for W:
2L + 2W = 540
2W = 540 - 2L
W = (540 - 2L) / 2
W = 270 - L
Now, substitute this value of W into the area formula A = L * W:
A = L * (270 - L)
To find the maximum area, we can differentiate this equation with respect to L and set it to zero:
dA/dL = 270 - 2L = 0
2L = 270
L = 135
Now that we have the value of L, we can substitute it back into the equation for W:
W = 270 - L
W = 270 - 135
W = 135
Therefore, the dimensions of the rectangle that will maximize the area are 135 feet by 135 feet.
Finally, calculate the area of each pen:
Each pen's area = (L/3) * (W/2) = (135/3) * (135/2) = 45 * 67.5 = 3037.5 square feet
Since there are six pens, the largest possible total area of the six pens would be:
Total area = 6 * 3037.5 = 18225 square feet.