Answer
30 and 30
Work Step by Step
The two numbers are $x$ and $60-x$. The product is $(x)(60-x)$, and we want this to be as large as possible. This product is as large as possible as the vertex.
$x*(60-x)$
$60x-x^2$
$y=-x^2+60x$
$a=-1$, $b=60$, $c=0$
Vertex: $x=-b/2a$
$x=-60/2*-1$
$x=-60/-2$
$x=30$
$x*(60-x)$
$30*(60-30)$
$30*30$
$900$
30 and 30