If you have 60 feet of fencing and want to enclose a rectangular area up against a long, straight wall, what is the largest area you can enclose?

Step 1 ：Let the length of the rectangle parallel to the wall be \(x\) and the width perpendicular to the wall be \(y\).

Step 2 ：Since we have 60 feet of fencing, we can write the equation: \(2y + x = 60\).

Step 3 ：Solve for \(y\) in the first equation: \(y = \frac{60 - x}{2}\).

Step 4 ：Find the area \(A\) by multiplying \(x\) and \(y\): \(A = x * y\).

Step 5 ：Substitute the expression for \(y\) into the area equation: \(A = x * \frac{60 - x}{2}\).

Step 6 ：Find the maximum area by taking the derivative of \(A\) with respect to \(x\) and setting it to zero: \(\frac{dA}{dx} = 30 - x\).

Step 7 ：Find the critical points: \(x = 30\).

Step 8 ：Find the corresponding value of \(y\): \(y = 15\).

Step 9 ：Calculate the maximum area: \(A = 30 * 15 = 450\).

Step 10 ：\(\boxed{450}\) square feet is the largest area that can be enclosed with 60 feet of fencing up against a long, straight wall.