# Suppose you have 200 feet of fencing to enclose a rectangular plot. How do you determine dimensions of the plot to enclose the maximum area possible?

The length and width should each be

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To determine the dimensions of the rectangular plot that will enclose the maximum area possible with 200 feet of fencing, follow these steps:

- Let (l) be the length and (w) be the width of the rectangular plot.
- The total length of fencing used will be (2l + 2w).
- Since the total length of fencing is given as 200 feet, we have (2l + 2w = 200).
- Rewrite the equation to solve for one variable in terms of the other: (l = 100 - w).
- The area ((A)) of the rectangular plot is given by (A = lw).
- Substitute the expression for (l) from step 4 into the equation for area: (A = (100 - w)w).
- Expand and simplify the expression: (A = 100w - w^2).
- To find the maximum area, take the derivative of the area function with respect to (w) and set it equal to zero: (\frac{dA}{dw} = 100 - 2w = 0).
- Solve for (w): (2w = 100 \Rightarrow w = 50).
- Substitute (w = 50) back into the equation for (l): (l = 100 - 50 = 50).
- So, the dimensions of the rectangular plot that enclose the maximum area possible with 200 feet of fencing are (50 \times 50) feet.

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