Bob has 20 feet of fencing to enclose a rectangular garden. If one side of the garden is x feet long, he wants the other side to be (10 – x) feet wide. What value of x will give the largest area, in square feet, for the garden?

Question

whpalmer4 · Answer

Area of the garden is $$A=\ell * w$$We know that one of those is $x$ and the other is $10-x$, so $$A = \ell * w = x(10-x) = 10x - x^2$$$$10x-x^2 = -x^2+10x$$which is a parabola in the form $ax^2 + bx + c$.  The coefficient $a$ of $-x^2$ is -1, so the parabola opens downward, and the vertex will be the maximum value of $A$ anywhere on the parabola.  You can find the x-value of the vertex of a parabola in that form by using $x = -b/2a$.  What is the value of $x$?  What are the respective lengths of the sides?  Do they add up to 20, as required?  Do you have any observations about the most efficient way to enclose an area if you pay for fencing by the foot?

whpalmer4 · Answer

So not as to completely give you the answer if you haven't finished the problem, here's a graph of the area function if Bob had 50 feet of fencing to work with.