A three sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region. There is 96 ft of fencing available. Find the maximum enclosed area and the dimensions of the corresponding enclosure.

I drew a picture of it and I got the area = x*y. I used x as the two widths and y as the length on the picture.

Well, it seems like you've done a great job so far! Let's see if we can figure out the dimensions and the maximum enclosed area.

Now, according to your picture, we have two sides of length x and one side of length y. The fourth side is formed by the river.

To enclose the rectangular region, we need a total of three sides, each with a length of x, and we also need the length of the fourth side, which is the river.

So, we know that:

3x + y = 96

Now, let's solve for y:

y = 96 - 3x

Substituting this value of y into our equation for the area, which is A = x * y, we get:

A = x * (96 - 3x)

To find the maximum enclosed area, we can take the derivative of A with respect to x and set it equal to zero:

dA/dx = 96 - 9x = 0

Solving for x, we have:

9x = 96
x = 96/9
x = 10.67 (approximately)

Since we can't have a fractional value for the length of a side, let's round x down to 10.

Now that we have x, we can substitute it back into our equation for y:

y = 96 - 3(10)
y = 96 - 30
y = 66

So, the dimensions of the corresponding enclosure are width = 10 ft, length = 66 ft, and the maximum enclosed area is A = 10 * 66 = 660 square feet.

Voila! You've found the maximum enclosed area and the dimensions of the corresponding enclosure. Hope that brings a smile to your face!

To find the maximum enclosed area, we need to determine the dimensions that will maximize the product of the lengths of the two sides, which is equivalent to maximizing the area (A = x * y).

Let's denote the length of the fence parallel to the river as x and the length perpendicular to the river as y.

Since the fence consists of three sides, we have the equation 2x + y = 96 ft (equation 1), which represents the total length of the fence.

To solve for x in terms of y, we can rearrange equation 1 as x = (96 - y) / 2.

Now, we substitute this expression for x into the formula for the area: A = x * y = ((96 - y) / 2) * y.

Expanding this equation, we have: A = (96y - y^2) / 2.

To find the maximum area, we take the derivative of A with respect to y and set it to zero, since the maximum occurs at the point of inflection.

dA/dy = (96 - 2y) / 2 = 0.

Simplifying the equation, we find 96 - 2y = 0.

Solving for y, we get y = 48.

Substituting this value back into equation 1, we find x = (96 - 48) / 2 = 24.

Therefore, the maximum enclosed area is A = x * y = 24 ft * 48 ft = 1152 ft^2, and the corresponding dimensions are 24 ft (width) by 48 ft (length).

To find the maximum enclosed area given the total length of fencing available, we can follow these steps:

1. Let's assume the length of the rectangular region parallel to the river is "x" and the width perpendicular to the river is "y". Therefore, the area of the rectangle is A = x * y.

2. We are given that the fencing available is 96 ft. Since we don't need to build a fence parallel to the river, we only need to consider the three sides perpendicular to the river. This means we have 2x + y = 96 ft.

3. Solve the equation from step 2 for y in terms of x: y = 96 - 2x.

4. Substitute the value of y from step 3 into the area equation from step 1: A = x * (96 - 2x).

5. Expand the equation to simplify: A = 96x - 2x^2.

6. The equation for the area A is now in quadratic form. To find the maximum value of A, we need to find the vertex of the quadratic equation. The vertex of a quadratic equation in the form of A = ax^2 + bx + c is given by x = -b/2a.

In this case, a = -2 and b = 96, so x = -96 / (2 * -2) = -96 / -4 = 24.

7. Since the dimensions of the rectangular region cannot be negative, discard the negative value of x. Therefore, x = 24 ft.

8. Substitute the value of x into the equation from step 2 to find y: y = 96 - 2(24) = 96 - 48 = 48 ft.

9. Therefore, the dimensions of the rectangular enclosure that maximizes the area are 24 ft by 48 ft, and the maximum enclosed area is A = 24 * 48 = 1152 sq ft.

so, now go for it. You know that

2x+y = 96, so y = 96-2x

the area is xy = 2x(96-2x) = 192x - 4x^2

That's just a parabola; its vertex gives the maximal area.

since there are usually four sides to a square(which is the optimal shape), you would do 96/4 to figure out what one side would be. Then since one side is missing, you'd add that one side to the other side. So s= side, 2s to find the other side length, which would then be 48. So the dimensions would just be, 24 by 24 by 48 to get the maximum area.