I need to express the radius $r$ of the right circular cone as a function of its height $h$ given that its volume equals to its lateral surface area. I know the two equations $\pi r \times \sqrt{r^2 + h^2}$ and the volume $\frac{\pi}{3} r^2 h$. Do I just set these equations as equal and solve for $h$? I'm not quite sure where to go from here.
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$\begingroup$The lateral surface area is $A_L = \frac{1}{2} C \times S$, where $C$ is the base circumference, and $S = \sqrt{r^2+h^2}$ is the distance from the tip of the cone to the point on the base circle.
Equality of the volume and the surface area gives you an equation: $$ \pi r \sqrt{r^2+h^2} = \frac{\pi}{3} r^2 h $$ Assuming $r>0$ and $h>0$, this simplifies to $3 \sqrt{r^2+h^2} = r h$. Squaring the left-hand-side and the right-hand-side will give the auxiliary equation, with the property that every solution of the original equation being a solution of the auxiliary equation. But the auxiliary equation might have extraneous solutions.
The auxiliary equation will be a simple quadratic equation, with two solutions. You should check which one of these will satisfy the original equation, and under which conditions this will be possible.
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