Setup Quadratic Word Problem

Question

I need help setting up this quadratic word problem, I have no idea where to start.

Among all pairs of (real) numbers whose sum is 17, find a pair whose product is as large as possible. What is the maximum product?

Answer 1

You know that there are two numbers whose sum is 17. Thus, for two numbers, $x$ and $y$, $$x + y = 17$$

Now we also want to find the maximum product, i.e. the greatest possible value of $xy$.

Now we know that $x = 17 - y$ (from above), and thus we just want to find the maximum value of $xy$, or $$(17-y)(y)$$

This is equivalent to $$17y-y^2$$

To find the maximum of this function, we can simply find when the derivative equals zero, or where $17-2y = 0$. The reason this works is because if you think about the shape of the function $17y-y^2$, we know it is a parabola that opens downwards, and so the only time the derivative will equal 0 is when the parabola is at its maximum point. Thus, if we solve for $17-2y=0$, we get $y = 8.5$.

Now, since $x+y=17$, and $y=8.5$, we know that $x=8.5$ also. Thus, the maximum value of $xy$ is when both $x$ and $y = 8.5$.

Thus, the maximum product is $8.5^2$.

Answer 2

Note the identity $$4xy=(x+y)^2-(x-y)^2.$$ If $x+y=17$, this gives $$4xy=17^2 -(x-y)^2.$$ The left side is as large as possible precisely if $(x-y)^2$ is as small as possible, namely $0$.

Thus the maximum value of $4xy$ is $17^2$, achieved when $x=y=\frac{17}{2}$.

The maximum value of $xy$ is therefore $\frac{17^2}{4}$.