Final answer:
To maximize the enclosed area with 208 feet of fencing for three sides of a rectangular lot, the optimal dimensions are a width of 52 feet and a length of 104 feet, giving an area of 5,408 square feet. Option 1 is the correct answer.
Step-by-step explanation:
For the developer who wants to enclose a rectangular area for parking with 208 feet of fencing, not including the side along the street, we must maximize the area within the constraints provided. Since the fence will cover only three sides of the rectangle (two widths and one length), the sum of these three sides must equal 208 feet.
Let's denote the length of the two equal sides (widths) as w and the single side parallel to the street (length) as l. The perimeter equation is therefore 2w + l = 208. To find the maximum area, we have to express the area as a function of one variable. The area A of a rectangle is A = w × l. Using the perimeter equation to express l in terms of w, we get l = 208 - 2w. Now we can express the area as A(w) = w(208 - 2w).
To find the maximum area, we differentiate A(w) with respect to w and set the derivative equal to zero to find the critical points. The critical point in this context yields w = 52 feet, which gives us l = 208 - 2(52) = 104 feet. Plugging these values back into the area formula gives us A = 52 × 104, which calculates to 5,408 square feet as the largest possible area that can be enclosed. The correct option is: O 10,816 ft.