Given 40 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the fencing?
I know that the area of a rectangle is length times width, and that the perimeter of a rectangle is 2 times the sum of the length and width. So, if I let L be the length and W be the width, I have two equations: A = LW and P = 2(L + W). I want to maximize A, but I also have a constraint on P: it has to be 40 feet. So, I can substitute P = 40 into the second equation and get 40 = 2(L + W), which simplifies to 20 = L + W. Now, I can use this equation to eliminate one of the variables in the first equation. For example, I can solve for W in terms of L: W = 20 - L. Then, I can plug that into the first equation and get A = L(20 - L). This is a quadratic function of L, and I can use calculus or algebra to find its maximum value. If I use calculus, I can take the derivative of A with respect to L and set it equal to zero: A' = 20 - 2L = 0. This gives me L = 10. Then, I can plug that back into the equation for W and get W = 10 as well. So, the maximum area occurs when L = W = 10, which means the rectangle is actually a square. The area of the square is 10 times 10, which is 100 square feet. 