Two positive numbers whose sum is 60 and the product of the square of…

Let the two positive numbers be $x$ and $y$, with $x + y = 60$.

Let the function to maximize be $P = x^2 y$.

Express $y$ in terms of $x$: $y = 60 - x$.

Then, $P = x^2 (60 - x) = 60x^2 - x^3$

Differentiating $P$ with respect to $x$:

$\frac{dP}{dx} = 120x - 3x^2$

Set $\frac{dP}{dx} = 0$:

$120x - 3x^2 = 0$

$x(120 - 3x) = 0$

$x = 0$ or $x = 40$

Since numbers are positive, $x = 40$

Then $y = 60 - 40 = 20$

Therefore, the two numbers are $40$ and $20$.