jmc

The cross section of the cone is an equilateral triangle. The ratio of the base to the height of an equilateral triangle is 1 to $\sqrt{3}/2$. In terms of the radius, $r$, the base is $2r$ and the height is $2r\sqrt{3}/2$, or $r\sqrt{3}$. Since we know the volume of the cone, we can use the volume formula and solve the equation $$(1/3)\times \pi \times r^2\times r\sqrt{3}=12288\pi$$ for $r$. Dividing both sides of the equation by $\pi$ gives $(1/3)r^3\sqrt{3}=12288$. Tripling both sides, we get $r^3\sqrt{3}=36,864$. Now, we want $r\sqrt{3},$ so we multiply both sides by $3$ to get $r^3\cdot (\sqrt{3}{)}^3=(r\sqrt{3}{)}^3=36,864\cdot 3=110,592.$ Taking the cube root of both sides, we get $r\sqrt{3}=\boxed{48.0}.$