Find two numbers whose sum is 34 and whose product is a maximum.

Answer 1

This problem appears to be the optimization of a system of equations.

Let x and y be our two numbers. We have two requirements for these two numbers:

1. x+y=34
2. x*y=M where M is some maximum

Let's rearrange the first equation to solve for y in terms of x:

y=34-x (subtract x from both sides)

Next, we will but this expression for y into our second equation and multiply out

x*(34-x)=M

34x-x²=M

Our x will be given by the peak (x,M) of this parabola, and the y value can then be found using y=34-x

The maximum and x values can be found either by graphing or by using the vertex equation, since we know the parabola 34x-x² is "downward facing", it's vertex will be the maximum.

For parabolas in ax²+bx+c form, the vertex point (x,M) can be found using x=-b/(2a), and then evaluating the formula for M.

In this case, a=-1 and b=34, so our x=(-34/-2)=17

y=34-17=17

M=17*17=289