Find the dimensions of the rectangular garden of greatest area that can be fenced off for $6000, if the cost of fencing is $5 per meter.

cost = 5 (2x+2y)

6000 = 10x + 10 y
x+y = 600
so y = 600-x
A =xy = 600x-x^2

dA/dx = 0 at max = 600 -2x
x = 300
then y = 300
so square

To find the dimensions of the rectangular garden with the greatest area that can be fenced off for $6000, let's break down the problem into smaller steps:

Step 1: Understand the problem:
We want to find the dimensions (length and width) of a rectangular garden. The cost of fencing is $5 per meter. We have a budget of $6000. We want to maximize the area of the garden.

Step 2: Define the variables:
Let L represent the length of the rectangular garden.
Let W represent the width of the rectangular garden.

Step 3: Set up the equations:
We know that the cost of fencing is $5 per meter, and we have a budget of $6000. Therefore, the total cost of fencing is given by the equation:
Cost of fencing = $5 * (perimeter of the garden)
$6000 = $5 * (2L + 2W)

We also want to maximize the area of the garden. The area of a rectangle is given by the equation:
Area = length * width
Area = L * W

Step 4: Solve the equations:
To find the dimensions of the garden with the maximum area, we need to find the values of L and W that satisfy both equations.

From the equation representing the cost of fencing, we can solve for L in terms of W:
2L + 2W = 6000 / 5
2L + 2W = 1200
L = (1200 - 2W) / 2
L = 600 - W

Now substitute this expression for L in the area equation:
Area = (600 - W) * W

Step 5: Maximize the area:
To maximize the area, we can differentiate the area equation with respect to W and set the derivative equal to 0. Then solve for W.

d(Area)/dW = (600 - W) - W = 0
600 - 2W = 0
600 = 2W
W = 300

Substitute this value of W back into the expression for L:
L = 600 - W
L = 600 - 300
L = 300

Step 6: Check the solution:
We need to confirm that the solution satisfies the budget constraint.
Cost of fencing = $5 * (2L + 2W)
Cost of fencing = $5 * (2*300 + 2*300)
Cost of fencing = $5 * (600 + 600)
Cost of fencing = $5 * 1200
Cost of fencing = $6000

The cost of fencing is indeed $6000, which satisfies the budget constraint.

Step 7: Final answer:
The dimensions of the rectangular garden of the greatest area that can be fenced off for $6000, at a cost of $5 per meter, are:
Length (L) = 300 meters
Width (W) = 300 meters