Question 3 of 14 Step 1 of 1 00:30:12 A farmer has 500 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 300 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.) Answer 8 Points Keypad Keyboard Shortcuts

Step 1 ：The problem is asking for the dimensions of the rectangle of maximum area that can be enclosed with 500 feet of fencing, using the barn as one side of the rectangle. The length of the barn is 300 feet.

Step 2 ：The area of a rectangle is given by the formula \(A = \text{length} \times \text{width}\). In this case, the length is fixed at 300 feet (the length of the barn), and the width is variable. However, the total amount of fencing available (500 feet) must be used to construct the other three sides of the rectangle.

Step 3 ：Therefore, the width of the rectangle (w) plus twice the length of the rectangle (2l) must equal 500 feet. Since the length is fixed at 300 feet, we can substitute this into the equation to get \(w + 2 \times 300 = 500\), or \(w = 500 - 2 \times 300\).

Step 4 ：We can then substitute this expression for w into the area formula to get \(A = 300 \times (500 - 2 \times 300)\). This is the function we need to maximize.

Step 5 ：To find the maximum of a function, we can take the derivative and set it equal to zero. However, in this case, the function is a simple quadratic, and we know that the maximum of a quadratic function occurs at its vertex. The x-coordinate of the vertex of a quadratic function given in the form \(f(x) = ax^2 + bx + c\) is \(-b/2a\). In this case, \(a = -2\) and \(b = 500\), so the maximum occurs at \(x = -500/(2 \times -2) = 125\).

Step 6 ：Therefore, the width of the rectangle that maximizes the area is 125 feet. The length is fixed at 300 feet.

Step 7 ：The result is negative, which is not possible in this context as the width of a rectangle cannot be negative. This suggests that there is an error in the calculation.

Step 8 ：Upon reviewing the problem, I realize that I made a mistake in the equation for the width of the rectangle. The equation should be \(w = (f - l) / 2\), not \(w = (f - 2 \times l) / 2\). This is because the total amount of fencing available (500 feet) must be used to construct the other three sides of the rectangle, not four.

Step 9 ：The result is now positive, which makes sense in this context. The width of the rectangle that maximizes the area is 100 feet, and the length is 300 feet. This is the final answer.

Step 10 ：Final Answer: The dimensions of the rectangle that maximize the area are length = 300 feet and width = \(\boxed{100}\) feet.