A ostrich farmer wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure below). He has 820 feet of fencing available to complete the job. What is the largest possible total area of the four pens?
To find the largest possible total area of the four pens, we need to determine the dimensions of the rectangular area first.
Let's assume the length of the rectangular area is L and the width is W. We are given that the farmer has 820 feet of fencing available.
The perimeter of the rectangular area is the sum of all the sides, which is equal to 2L + W (since there are two lengths and one width).
Given that the perimeter is equal to 820 feet, we have the equation:
2L + W = 820
We want to maximize the total area of the four pens, which is equal to L * W. Since L * W is a product, to maximize the area, we need to maximize both L and W.
Now let's solve the equation for W:
2L + W = 820
W = 820 - 2L
Substitute this value of W in the area equation:
Total area = L * (820 - 2L)
To find the maximum area, we can find the vertex of the parabola by taking the derivative of the area equation and setting it equal to zero:
d(area)/dL = 820 - 4L = 0
820 = 4L
L = 205
Substituting this back into the equation for W:
W = 820 - 2L
W = 820 - 2(205)
W = 410
So, the dimensions of the rectangular area that maximize the total area of the four pens is 205 feet by 410 feet.
Now, to find the largest possible total area, we calculate L * W:
Total area = 205 * 410
Total area = 84,050 square feet
Therefore, the largest possible total area of the four pens is 84,050 square feet.