A farmer has 460 feet of fencing with which to enclose a rectangular grazing pen next to a barn. The farmer will use the barn as one side of the pen, and will use the fencing for the other three sides. find the dimension of the pen with the maximum area?

L + 2 W = 460

so
L = (460 - 2 W)

L * W = A
A = (460-2W)W = 460 W - 2 W^2
DA/DW = 460 - 4 W
ZERO FOR MAX OR MIN
4 W = 460
W = 115

L = 460 - 2*115 = 230

To find the dimensions of the pen with the maximum area, we can use the fact that the area of a rectangle is given by the equation A = length * width.

Let's assume the length of the pen is L and the width of the pen is W.

We know that the farmer will use the barn as one side of the pen, which means we only need to fence off the remaining three sides. Therefore, the total length of fencing required will be equal to the perimeter of the pen, given by P = 2L + W.

According to the problem, the farmer has 460 feet of fencing, so we have the equation:

2L + W = 460

We can solve this equation for W:

W = 460 - 2L

Now, let's substitute this value of W into the area equation:

A = L * W

A = L * (460 - 2L)

Expanding the equation:

A = 460L - 2L^2

To find the maximum area, we need to find the value of L that maximizes the area A. We can do this by finding the vertex of the quadratic equation -2L^2 + 460L.

The x-coordinate of the vertex is given by:

L = -b / (2a)

For our equation, a = -2 and b = 460, so plugging these values in:

L = -460 / (2(-2))
L = 460 / 4
L = 115

We substitute this value of L back into the equation to find the corresponding width:

W = 460 - 2L
W = 460 - 2(115)
W = 460 - 230
W = 230

Therefore, the dimensions of the pen with the maximum area are 115 feet in length and 230 feet in width.

To find the dimensions of the pen with the maximum area, we need to determine the length and width of the rectangular grazing pen.

Let's assume the length of the pen is L and the width of the pen is W. Since the barn is one side of the pen, we can visualize the pen as a rectangle adjacent to the barn. So, we only need to calculate the length and width of the remaining three sides.

The perimeter of the pen is given by the sum of the lengths of the three sides:

Perimeter = 2L + W

According to the problem, the farmer has 460 feet of fencing available, so we have:

2L + W = 460

Now, we need to express the area of the pen in terms of either L or W. The area of a rectangle is calculated by multiplying its length by its width:

Area = L × W

To find the maximum area, we can express one of the variables, either L or W, in terms of the other variable using the perimeter equation.

From the perimeter equation, we can rearrange it to express one of the variables in terms of the other. Let's solve for L:

2L = 460 - W

L = (460 - W) / 2

Now, substitute this expression for L in the area equation:

Area = ((460 - W) / 2) × W

To find the maximum area, we need to find the value of W that maximizes this area. We can do this by finding the vertex of the quadratic equation representing the area.

The area equation can be simplified as follows:

Area = (230W - 0.5W^2)

This equation represents a downward-opening parabola. To find the vertex, we can take the derivative of the area equation and set it equal to zero:

d(Area) / dW = 230 - W = 0

Solving for W, we find:

W = 230

So, the width that maximizes the area is 230 feet.

Now substitute this value of W back into the perimeter equation to find the length:

2L + 230 = 460

2L = 460 - 230

2L = 230

L = 115

Therefore, the dimensions of the pen with the maximum area are 115 feet in length and 230 feet in width.

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