The surface area of a solid cylinder is found using the formula $SA = 2\pi r^2 + 2\pi rh$. The volume of a cone is found using the formula $V = \frac{1}{3} \pi r^2 h$. Judy has a solid cylinder whose total surface area is numerically equal to the volume of a particular cone that has a base congruent to the cylinder's base. If the height of both solids is $4$ inches, what is the radius?
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Setting the surface area of the cylinder equal to the volume of the cone, we have \[2\pi r^2 + 2\pi rh = \frac13 \pi r^2 h.\]We divide by $\pi r$: \[2r + 2h = \frac13 r h.\](That is, we divide by $\pi r^2$ and multiply by $3$ to cancel the factor of $\frac13 \pi r^2 h$ on the right side.) Substituting $h = 4,$ we have \[2r + 2(4) = \frac13 r(4).\]Simplifying the equation gives $2r + 8 = \frac{4}{3}r,$ so $\frac{2}{3}r = 8,$ or $r = \boxed{12}.$
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