a farmer wants to fence in part of her land so that her chickens will have thier own little area. if she only has 28 feet of fence, what is the area of the largest pen that she can build?
To find the area of the largest pen that the farmer can build, we need to determine the dimensions of the pen that would maximize its area.
Let's assume that the rectangular pen has one side parallel to the existing fences and the other side perpendicular to them. Let's call the length of the side parallel to the fences "L" and the length of the side perpendicular to the fences "W."
Since the perimeter of a rectangle is given by the formula P = 2L + 2W, and the farmer has only 28 feet of fence available, we can write the equation:
2L + 2W = 28
Now, we need to express the area "A" of the pen in terms of L and W. The area of a rectangle is given by the formula A = L * W.
To optimize the area, we can solve for one variable in terms of the other using the perimeter equation and substitute it into the area equation. Let's solve the perimeter equation for L:
2L + 2W = 28
2L = 28 - 2W
L = 14 - W
Now substitute L into the area equation:
A = L * W
A = (14 - W) * W
A = 14W - W^2
To find the maximum area, we can take the derivative of A with respect to W and set it equal to zero:
dA/dW = 14 - 2W = 0
Solving for W:
14 - 2W = 0
2W = 14
W = 7
Now substitute this value of W back into the equation for L:
L = 14 - W
L = 14 - 7
L = 7
Thus, the dimensions of the pen that would maximize its area are 7 feet by 7 feet. To find the maximum area, substitute these values back into the area equation:
A = L * W
A = 7 * 7
A = 49 square feet
Therefore, the area of the largest pen that the farmer can build with 28 feet of fence is 49 square feet.
P = 2L + 2W
28/4 = 7
7 by 7
6 by 8
5 by 9
Which of those measurements would give you the largest area?
A = LW