Final answer:
The maximum area of a rectangular playground with 5n feet of fencing is 25n^2/16.
Step-by-step explanation:
The maximum area of a rectangular playground that can be fenced with 5n feet of fencing is achieved when the sides of the rectangle are equal. Let's call the length and width of the rectangle 'l' and 'w' respectively.
Since each side of the rectangle will be fenced, the perimeter of the rectangle is 5n, which is equal to 2l + 2w. Rearranging this equation, we get l + w = 5n/2.
To find the maximum area, we need to maximize the product of length and width. To do this, we can use the equation l = 5n/2 - w and substitute it into the equation for the area of a rectangle, A = l * w. Simplifying, we have A = (5n/2 - w) * w = (5n/2)w - w^2.
To find the maximum area, we can differentiate A with respect to w, set the derivative equal to zero, and solve for w. Taking the derivative of A, we get dA/dw = 5n/2 - 2w. Setting this derivative equal to zero, we have 5n/2 - 2w = 0. Solving for w, we find w = 5n/4.
Substituting this value of w back into the equation for the area, we have A = (5n/2)(5n/4) - (5n/4)^2 = 25n^2/8 - 25n^2/16 = 25n^2/16. Therefore, the maximum area of the playground in terms of n is 25n^2/16.