A farmer wishes to fence in 3 different breeds of animals in a rectangular area and keep all the breeds in separate areas. If the farmer has 144 feet of fence what is the maximum area he can fence in? Type in your answer to the nearest whole number
To find the maximum area that the farmer can fence in, we need to determine the dimensions of the rectangular area. Let's assume the length of the rectangular area is L and the width is W.
Since the farmer wants to keep the three different breeds of animals in separate areas, we can divide the rectangular area into three equal parts. This means that each breed will have its own section with the same width, W, and one-third of the total length, L/3.
To calculate the amount of fence needed, we add up the lengths of the sides: 2L + 4W. We know that the farmer has 144 feet of fence, so we can write the following equation:
2L + 4W = 144
To maximize the area, we need to express the area in terms of a single variable. The formula for the area of a rectangle is A = L * W.
Substituting L/3 for L, we get:
A = (L/3) * W
Now, we can solve for one variable in terms of the other. Let's express L in terms of W:
2L + 4W = 144
2(L/3) + 4W = 144
2L + 12W = 432
2L = 432 - 12W
L = (432 - 12W) / 2
L = 216 - 6W
Now substitute this expression for L into the area formula:
A = (L/3) * W
A = ((216 - 6W)/3) * W
A = (72 - 2W) * W
A = 72W - 2W^2
To maximize the area, we take the derivative of A with respect to W and set it equal to zero:
dA/dW = 72 - 4W = 0
Now solve for W:
72 - 4W = 0
4W = 72
W = 18
We now have the value of W. To find the corresponding value of L, we use the equation we derived earlier:
L = 216 - 6W
L = 216 - 6(18)
L = 216 - 108
L = 108
So the dimensions of the rectangular area that maximize the area and use all 144 feet of fencing are:
Length (L) = 108 feet
Width (W) = 18 feet
To find the maximum area, substitute these dimensions into the formula:
A = L * W
A = 108 * 18
A = 1944
Therefore, the maximum area that the farmer can fence in is approximately 1944 square feet.