You have 192 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. Find the largest area that can be covered and the length of each side

192 / 3 = 64

A= 64 * 64

A = 4,096 square feet

To find the dimensions that will maximize the area, we can use the concept of derivatives from calculus. Let's denote the length of the plot as L and the width as W.

Given that we have 192 feet of fencing, we know that the perimeter of the plot must equal 192 feet. The perimeter is calculated as follows:

P = 2L + W

However, since we're not fencing the side along the river, the perimeter equation becomes:

P = L + 2W

Substituting P with 192, we have:

192 = L + 2W

We can now isolate L in terms of W:

L = 192 - 2W

Now, the area of the plot is given by multiplying the length and width:

A = L * W

Substituting L with 192 - 2W:

A = (192 - 2W) * W

Expanding the equation, we get:

A = 192W - 2W^2

To find the values of W that maximize the area, we need to take the derivative of A with respect to W and set it equal to zero. Let's find the derivative:

dA/dW = 192 - 4W

Setting it equal to zero:

192 - 4W = 0

Solving for W, we have:

4W = 192
W = 48

Now we can substitute W back into the equation for L:

L = 192 - 2W
L = 192 - 2(48)
L = 96

So, the width (W) that maximizes the area is 48 feet, and the length (L) is 96 feet.

To find the maximum area (A), we substitute the values of L and W into the area equation:

A = L * W
A = 96 * 48
A = 4608 square feet

To determine the length and width of the plot that will maximize the area, we can use the concept of optimization.

Let's denote the length of the plot as "L" and the width as "W". Since we are not fencing the side along the river, we have two equal lengths of fencing on the opposite sides of the rectangular plot, and one width of fencing. So, we can set up the equation for the perimeter as follows:

Perimeter = 2L + W = 192

We want to maximize the area, which is given by the formula:

Area = L * W

To solve this problem, we can express one of the variables in terms of the other and substitute it into the area formula, and then differentiate the area with respect to the remaining variable to find its critical points.

Let's express the width "W" in terms of the length "L" using the perimeter equation:

W = 192 - 2L

Now we can substitute this expression for width in terms of length into the area formula:

Area = L * (192 - 2L) = 192L - 2L^2

To find the critical points, we need to differentiate the area equation with respect to "L" and set it equal to zero:

d(Area)/dL = 192 - 4L = 0

Now let's solve this equation to find the value of "L":

192 - 4L = 0
4L = 192
L = 48

So, the length of the plot is 48 feet.

Now we can substitute this value back into the width equation to find the width:

W = 192 - 2L
W = 192 - 2(48)
W = 192 - 96
W = 96

Therefore, the width of the plot is 96 feet.

To obtain the largest area, we can substitute the found values of length and width into the area formula:

Area = L * W
Area = 48 * 96
Area = 4608

Hence, the largest area that can be covered is 4608 square feet, with the length of each side being 48 feet.