Answer: Let's assume the width of the rectangular plot is represented by "x" feet, and the length is represented by "y" feet.
The cost of heavy-duty fencing along the width is $4.5 per foot, so the cost for the width fencing would be 4.5x dollars.
The cost of standard fencing along the length is $2.5 per foot, so the cost for the length fencing would be 2.5y dollars.
The total cost of fencing is given as $6000. So we can write the equation:
4.5x + 2.5y = 6000
To find the dimensions of the rectangular plot of the greatest area, we need to maximize the area. The area of a rectangle is given by the formula A = length × width, which in this case is A = xy.
We need to maximize the area A, subject to the constraint equation 4.5x + 2.5y = 6000.
To solve this problem, we can use the method of Lagrange multipliers or substitution. Here, we'll use substitution.
Solve the constraint equation for y:
2.5y = 6000 - 4.5x
y = (6000 - 4.5x) / 2.5
Now substitute the value of y in the area formula:
A = x * [(6000 - 4.5x) / 2.5]
Simplify the equation:
A = (3/5)x(6000 - 4.5x)
A = (3/5)(6000x - 4.5x^2)
To find the maximum area, we differentiate the equation with respect to x, set it equal to zero, and solve for x.
dA/dx = 0
(3/5)(6000 - 4.5x) - (3/5)(4.5x) = 0
(3/5)(6000 - 9x) = 0
6000 - 9x = 0
9x = 6000
x = 6000/9
x ≈ 666.67
Substitute the value of x back into the equation for y:
y = (6000 - 4.5x) / 2.5
y = (6000 - 4.5(666.67)) / 2.5
y ≈ 799.99
So, the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000 are approximately:
Width (x) ≈ 666.67 feet
Length (y) ≈ 799.99 feet