a farmer has 400 meters of fence with which to enclose a portion of land. the farmer wants to enclose a rectangular piece of ground that is as large as possible. the land is bordered by water on two sides.

There are three options for the farmer
Option 1) have two sides bordered by the water and two sides by the fence
Option 2) have one side bordered by water and three sides by the fence
Option 3) use fence for all four sides

Construct formula for the area in each of the three options( hint: name one side x)

I don't under stand how to setup the formula besides setting each one by "x=..."

(1)

The sides are x and y, so y = 400-x
The area is thus a = xy = x(400-x)

(2)
If the side x is parallel to the water,
x+2y = 400, so y = (400-x)/2 and the area is
a = xy = x*(400-x)/2

(3)
x+y=200, so the area is
a = xy = x(200-x)

If you need to find the maximum area, just remember that the graph of a is a parabola, with max area at the vertex. So, find the vertex of each parabola.

I still don't understand

OK. You do understand that if the sides of the rectangle are x and y, then the area is

a = xy

right?

Given the equation of a parabola, can you find the vertex?

If y = ax^2+bx+c, then the vertex is at x = -b/2a

So, in each case, figure out what x and y are, express the area a as a quadratic function of x alone.

So, what in particular don't you get?

To set up the formulas for the area in each of the three options, let's first understand the dimensions of the rectangular piece of ground in each case.

Option 1: Two sides bordered by water and two sides by the fence
In this case, the length of the rectangle will be the total length of the fence, as the two sides bordered by water do not need fence. The width of the rectangle will be x, as we need to name one side x.

Therefore, the formula for the area (A) in option 1 would be:
A = length x width
A = (400 - 2x) x x
A = 400x - 2x^2

Option 2: One side bordered by water and three sides by the fence
In this case, one side of the rectangle will be the total length of the fence minus x, as one side is bordered by water. The other two sides will be x.

Therefore, the formula for the area in option 2 would be:
A = (length - x) x width
A = (400 - x) x x
A = 400x - x^2

Option 3: Use fence for all four sides
In this case, all four sides will be made up of the fence. Since we need to name one side as x, both length and width will be x.

Therefore, the formula for the area in option 3 would be:
A = length x width
A = x x x
A = x^2

So, the formulas for the area in each of the three options are:
- Option 1: A = 400x - 2x^2
- Option 2: A = 400x - x^2
- Option 3: A = x^2

Now, these formulas represent the different areas possible with the given constraints. By substituting different values of x into the formulas, you can find the maximum area achievable in each option.