a farmer has 400 meters of fence with which to enclose a portion of land. the farmer wants to enclose a rect angular 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)

All I have is ¨x=...¨ Someone please help

(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.

To find the formula for the area in each of the three options, we can break down the problem and analyze each scenario separately.

Option 1:
In this option, two sides are bordered by water and two sides are bordered by the fence. Let's name the length of the two sides bordered by water as x and the other two sides (bordered by the fence) as y.

Since the total length of the fence is 400 meters, we have the equation:
2x + y = 400

To find the area, we multiply the length of the two sides:
Area = x * y

Option 2:
In this option, one side is bordered by water and three sides are bordered by the fence. Let's name the length of the side bordered by water as x and the other three sides (bordered by the fence) as y.

Again, based on the total length of the fence, we have the equation:
x + 3y = 400

To find the area, we multiply the length of the two sides:
Area = x * y

Option 3:
In this option, all four sides are made up of the fence. Let's name the length of each side as x.

Since all four sides are made up of the fence, we have the equation:
4x = 400

To find the area, we multiply the length of the two sides:
Area = x * x = x^2

Now that we have the equations for each option, we can substitute the values into the formulas to get the expressions for the areas:

Option 1:
Area = x * y

Option 2:
Area = x * y

Option 3:
Area = x^2

Remember, these formulas represent the area in terms of the variable x.

To construct the formulas for the area in each of the three options, let's consider the given information and find the equations step-by-step:

Option 1: Two sides bordered by water and two sides by the fence.
Since two sides are bordered by water, they are not enclosed by the fence, leaving two sides enclosed by the fence. Let's assume the side adjacent to the water has a length of x meters. Since the opposite side will also have a length of x meters, we can represent the dimensions of the rectangular enclosure as x by x.

Therefore, the area of the enclosed land in Option 1 is:
Area = x * x = x^2

Option 2: One side bordered by water and three sides by the fence.
Let's assume the side adjacent to the water has a length of x meters, and the other two sides enclosed by the fence are of lengths y meters and z meters. Since one side is adjacent to the water, we can assume that the lengths y and z are equal. Therefore, the dimensions of the rectangular enclosure can be represented as x by y by y.

Therefore, the area of the enclosed land in Option 2 is:
Area = x * y * y = x * y^2

Option 3: Use fence for all four sides.
In this case, all four sides of the rectangular enclosure will be enclosed by the fence. Let's assume the dimensions of the rectangular enclosure are a meters and b meters. Therefore, the dimensions of the rectangular enclosure can be represented as a by b.

Therefore, the area of the enclosed land in Option 3 is:
Area = a * b

To summarize, the formulas for the area in each of the three options are:
Option 1: Area = x^2
Option 2: Area = x * y^2
Option 3: Area = a * b