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