Problem

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

Solution

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

From Solvely APP
Source: https://solvelyapp.com/problems/8369/

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