Question 10 (10 points)

You have $800 \mathrm{ft}$ of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

Step 1 ：The problem is asking for the dimensions of a rectangular pen that maximizes the area, given that one side of the pen is a river and we have 800 ft of fencing. This is a problem of optimization.

Step 2 ：We know that the area of a rectangle is given by the formula \(A = \text{length} \times \text{width}\). Since one side of the rectangle is a river, we only need fencing for the other three sides. Let's denote the length of the rectangle (parallel to the river) as \(x\) and the width (perpendicular to the river) as \(y\).

Step 3 ：We know that the total length of the fencing is 800 ft, so we have the equation \(x + 2y = 800\). We can solve this equation for \(y\) to get \(y = (800 - x) / 2\).

Step 4 ：We can substitute this into the area formula to get \(A = x \times ((800 - x) / 2)\). This is a quadratic function, and we know that the maximum value 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, \(a = -1/2\) and \(b = 800\), so the x-coordinate of the vertex is \(-800 / (2 \times -1/2) = 800\).

Step 5 ：We can substitute \(x = 800\) into the equation for \(y\) to get the corresponding y-coordinate.

Step 6 ：Let's calculate these values.

Step 7 ：\(x = 400.0\)

Step 8 ：\(y = 200.0\)

Step 9 ：Final Answer: The dimensions of the rectangular pen that maximize the area are \(\boxed{400 \, \text{ft}}\) for the length and \(\boxed{200 \, \text{ft}}\) for the width.