jmc

By AM-GM, $$60=4x+9y\ge 2\sqrt{(4x)(9y)}=2\sqrt{36xy}=12\sqrt{xy},$$so $\sqrt{xy}\le 5.$ Hence, $xy\le 25.$

Equality occurs when $4x=9y.$ Along with the condition $4x+9y=60,$ we can solve to get $x=\frac{15}{2}$ and $y=\frac{10}{3},$ so the maximum value is $\boxed{25}.$