A farmer has 520520 feet of fencing to construct a rectangular pen up against the straight side of a barn, using the barn for one side of the pen. The length of the barn is 310310 feet. Determine the dimensions of the rectangle of maximum area that can be enclosed under these conditions. (Hint: Be mindful of the domain of the function you are maximizing.)

Sure, let's solve this step by step:

Step 1: Understand the problem The farmer has 520 feet of fencing and wants to construct a rectangular pen using the barn as one side. The barn is 310 feet long. We need to find the dimensions of the rectangle that will give the maximum area.

Step 2: Define the variables Let's denote the length of the rectangle (parallel to the barn) as L and the width (perpendicular to the barn) as W. Since the barn is one side of the rectangle, we only need fencing for the other three sides. Therefore, the total length of the fencing is L + 2W = 520 feet.

Step 3: Write the equation for the area The area of a rectangle is given by the product of its length and width, so A = L * W.

Step 4: Express W in terms of L From the equation in step 2, we can express W as W = (520 - L) / 2.

Step 5: Substitute W in the area equation Substituting W in the area equation gives A = L * (520 - L) / 2.

Step 6: Find the maximum area To find the maximum area, we need to find the derivative of A with respect to L, set it equal to zero, and solve for L. The derivative of A is A' = 520/2 - L = 260 - L. Setting this equal to zero gives L = 260 feet.

Step 7: Find W Substituting L = 260 feet into the equation from step 4 gives W = (520 - 260) / 2 = 130 feet.

So, the dimensions that will give the maximum area are a length of 260 feet and a width of 130 feet.