The maximum area of the rectangular garden, obtained by solving the optimization problem, is 625/16 square feet.
Certainly! Let's solve the optimization problem step by step:
Given:
2l + 2w = 25
We can solve for l in terms of w:
l + w = 25/2
l = 25/2 - w
Now, express the area A in terms of w:
A = lw
A = (25w/2 - w^2)
Expand and simplify the equation:
A = 25w/2 - w^2
To find the critical points, take the derivative of A with respect to w and set it equal to zero:
dA/dw = 25/2 - 2w
Setting dA/dw = 0:
25/2 - 2w = 0
-2w = -25/2
w = 25/4
Now, substitute w = 25/4 back into the equation for l:
l = 25/2 - 25/4
l = 25/4
So, the dimensions that maximize the area are l = 25/4 and w = 25/4.
Finally, calculate the maximum area:
A_max = (25/4) * (25/4)
A_max = 625/16
Therefore, the maximum area of the rectangular garden is 625/16 square feet.