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 $6$ inches, what is the radius?
1 answer
We are given the equation $2\pi r^2 + 2\pi rh = \frac{1}{3} \pi r^2 h$, but since $h=6$, $2\pi r^2 + 12\pi r = \frac{1}{3} \pi r^2 6$. Dividing by $\pi r$, we have $2r+12 = \frac{1}{3} r \cdot 6$. Solving this equation gives $r = \boxed{18}$.