The back of Dante's property is a creek. Dante would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture. If there is 720 feet of fencing available, what is the maximum possible area of the pasture?

64800

A square would get the maximum area.

(720/3)^2 = ?

Well, Dante certainly has a tricky situation on his hands. He wants to create the maximum possible area of pasture with only 720 feet of fencing. Let's put on our thinking caps and get started, shall we?

Since the creek acts as one side of the pasture, Dante needs to fence in the other three sides. Let's call the length of the pasture L and the width W.

Now, we know that the length and the width should both consume some part of the 720 feet of fencing available. So, the total length of fencing used would be 2W (the two width sides) plus L (the length side).

We can create an equation to represent this: 2W + L = 720.

Now, to maximize the area, we can express the area of the pasture as A = L * W.

To maximize A, let's solve for one variable in terms of the other. We can rearrange the first equation to give: L = 720 - 2W.

Substituting this value of L into the equation for A, we get A = (720 - 2W) * W.

To find the maximum area, we will differentiate the equation for A with respect to W, set it equal to zero, and solve for W.

Here comes the punchline... Once we solve the equation, we find that W = 180. So, the width of the pasture should be 180 feet.

Now that we have the width, we can substitute it back into the equation for L to find its value: L = 720 - 2(180) = 360 feet.

Finally, we can calculate the maximum area by multiplying the length and width: A = 360 * 180 = 64,800 square feet.

So, drumroll please... the maximum possible area of Dante's pasture is 64,800 square feet! I hope Dante's cows will have plenty of space to graze and enjoy themselves.

To find the maximum possible area of the pasture, we need to determine the dimensions of the rectangular area that would allow us to enclose the largest possible area using the available fencing.

Let's assume that the length of the rectangular area is parallel to the creek, so the creek forms one of the sides. Let's call the length of the rectangular area L (in feet) and the width W (in feet).

Since the creek forms one side, we only need to enclose three sides with fencing. The two equal-length sides will be parallel to the width, and the remaining side will be parallel to the length.

Therefore, we can write the equation for the total length of the fencing used as:
Length of fencing = 2W + L

Given that there are 720 feet of fencing available, we can write the equation as:
720 = 2W + L

To solve for L in terms of W, we can rearrange the equation to:
L = 720 - 2W

Now, we need to find the maximum possible area of the rectangular area. The area (A) of a rectangle is calculated by multiplying the length (L) and the width (W):
A = L * W

Substituting the expression for L, we get:
A = (720 - 2W) * W

To find the maximum possible area, we need to find the value of W that maximizes the area A. We can do this by finding the vertex of the parabola formed by the equation.

The formula for the x-coordinate of the vertex of a quadratic equation in standard form (ax^2 + bx + c) is given by:
x = -b / (2a)

In our equation, the coefficient of the quadratic term is -2, and there is no x-term.
Therefore, the x-coordinate of the vertex is given by:
W = -(-2) / (2 * 1)

Simplifying:
W = 2 / 2
W = 1

Now that we have the value of W, we can substitute it back into the expression for L:
L = 720 - 2(1)
L = 720 - 2
L = 718

Therefore, the dimensions of the rectangular area that would maximize the area are W = 1 foot and L = 718 feet.

To find the maximum possible area, we can substitute these values into the equation for A:
A = L * W
A = 718 * 1
A = 718 square feet

Therefore, the maximum possible area of the pasture is 718 square feet.

57600