A solid brass cube with height $h$ is melted and recast as a solid cone of height $h$ and base radius $r$. We need to find $r$ in terms of $h$.

1. Problem Description

A solid brass cube with height $h$ is melted and recast as a solid cone of height $h$ and base radius $r$. We need to find $r$ in terms of $h$.

2. Solution Steps

The volume of the cube is $V_{cube} = h^3$.

The volume of the cone is $V_{cone} = \frac{1}{3} \pi r^2 h$.

Since the cube is melted and recast as a cone, their volumes are equal:

$V_{cube} = V_{cone}$

$h^3 = \frac{1}{3} \pi r^2 h$

Divide both sides by $h$:

$h^2 = \frac{1}{3} \pi r^2$

Multiply both sides by 3:

$3h^2 = \pi r^2$

Divide both sides by $\pi$:

$\frac{3h^2}{\pi} = r^2$

Take the square root of both sides:

$r = \sqrt{\frac{3h^2}{\pi}}$

$r = h \sqrt{\frac{3}{\pi}}$

3. Final Answer

D. $r = h \sqrt{\frac{3}{\pi}}$