Question 873702
{{{x}}}= one of the numbers.
{{{700-x}}}= the other number.
{{{y=x(700-x)=700x-x^2}}}= the product of those two numbers.
That is a quadratic function, and is usually written as
{{{y=-x^2+700x}}} .
The graph is a parabola, looking like this: {{{graph(300,300,-200,800,-50000,150000,-x^2+700x)}}} .
Where is its maximum?
You may have been shown, and it can be proven that the maximum of a quadratic like
{{{y=ax^2+bx+c}}} is at {{{x=-b/"2 a"}}}
So in this case the maximum happens when
{{{x=(-700)/(2*(-1))=(-700)/(-2)=highlight(350)}}} .
Of course, that means that the other number is
{{{700-x=700-350=highlight(350)}}} .
That is not surprising because it is the same as asking you what rectangle with a half-perimeter of 700 has the greatest area, and we know it is a square.
The same problem could also be asked as what is the largest rectangular plot that can be fenced when you have 1400 feet of fencing.