30. You have 64 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area.
A length: 48 feet, width: 16 feet
B. length: 32 feel, width: 16 feet
C. length: 32 feet, width: 32 feet
D. length: 16 feet, width: 16 feet

Step 1 ：Given 64 feet of fencing, let the length of the plot be L and the width be W. The equation for the fencing is: \(2W + L = 64\)

Step 2 ：We want to maximize the area A, which is given by: \(A = L * W\)

Step 3 ：Solve for L in the fencing equation: \(L = 64 - 2W\)

Step 4 ：Substitute L into the area equation: \(A(W) = (64 - 2W) * W\)

Step 5 ：Find the maximum area by taking the derivative of A(W) with respect to W and setting it to zero: \(dA/dW = 64 - 4W\)

Step 6 ：Solve for W: \(0 = 64 - 4W\) => \(W = 16\)

Step 7 ：Find the length L using the equation: \(L = 64 - 2W\) => \(L = 64 - 2(16)\) => \(L = 32\)

Step 8 ：\(\boxed{\text{Final Answer: The length and width of the plot that will maximize the area are 32 feet and 16 feet}}\)