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A rectangular lot is to be bounded by a fence. Two kinds of fencing will be used with heavy duty fencing selling for $4.5 per foot along the width, and standard fencing selling for $2.5 per foot along the length. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000 ? Use two decimal places, if necessary, in your calculations. Round your final answers to two decimal places, if necessary, and write out units.

User Mirel
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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

User OSH
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