if you have 88 ft of fencing to enclose a rectangular plot but don't fence one side, find the length and width of the rectangle that will maximize the area
To find the length and width of the rectangle that will maximize the area, we need to use optimization techniques. Here's the step-by-step process:
1. Let's assume the length of the rectangle is x ft and the width is y ft.
2. The perimeter of the rectangle is given by the sum of all sides: 2x + y.
3. According to the problem, we have 88 ft of fencing available to enclose the three sides (2x + y). Therefore, the equation becomes: 2x + y = 88.
4. Now, we need an equation representing the area of the rectangle. The area of a rectangle is given by the product of its length and width: A = xy.
5. We need to rewrite the equation from step 3 in terms of one variable to substitute it into the equation for the area. Solving the equation for y, we get: y = 88 - 2x.
6. Substituting y in terms of x into the area equation, we have: A = x(88 - 2x) = 88x - 2x^2.
7. To maximize the area, we need to find the critical points of the equation. Taking the derivative of A with respect to x and setting it equal to zero, we get: dA/dx = 88 - 4x = 0.
8. Solving the equation 88 - 4x = 0 yields: x = 22.
9. Now, substitute x = 22 back into 2x + y = 88 to find y: 2(22) + y = 88. Simplifying the equation gives: y = 44.
10. Therefore, the length of the rectangle that will maximize the area is 22 ft, while the width is 44 ft.
To summarize, to maximize the area with 88 ft of fencing to enclose a rectangular plot without fencing one side, the length of the rectangle should be 22 ft, and the width should be 44 ft.
Area=l*w
88=l+2w
Area=w(88-2w)=-2w^2 +88w
Graph y=-2w^2 + 88w and see where the max is.