## Task 17/69

For a right circular cone, let the surface area $A$ and the ratio $k$ of the height $h$ to the radius $r$ be given $\left(k=\frac{h}{r}\right)$.

We are looking for a formula that expresses the volume $V$ in terms of $A$ and $k$.
In the common formula $V=\frac{\pi r^{2} h}{3}$, $r$ and $h$ are to be expressed in terms of $A$ and $k$. From

$$
\begin{gathered}
A=\pi r\left(r+\sqrt{r^{2}+h^{2}}\right) \quad \text { and } \quad h=k r \quad \text { we get } \\
r^{2}=\frac{A}{\pi\left(1+\sqrt{1+k^{2}}\right)} \quad \text { and } h=k \sqrt{\frac{A}{\pi\left(1+\sqrt{1+k^{2}}\right)}}
\end{gathered}
$$

and thus the desired formula

$$
V=\frac{A k}{3\left(1+\sqrt{1+k^{2}}\right)} \sqrt{\frac{A}{\pi\left(1+\sqrt{1+k^{2}}\right)}}
$$