7. A cone and a cylinder, with their bottom faces on the same plane, have a common inscribed sphere. Let the volume of the cone be $V_{1}$, and the volume of the cylinder be $V_{2}$, and $V_{1}=k V_{2}$, then $k_{\text {min }}=$ $\qquad$
7. $\frac{4}{3}$ Detailed explanation: Let the radius of the inscribed sphere be $R$, then the volume of the cylinder is $2 \pi R^{3}$. Let the base angle of the axial section of the cone be $2 \alpha$, the volume of the cone can be calculated as $\frac{2}{3} \pi R^{3} \cdot \frac{1}{\tan ^{2} \alpha\left(1-\tan ^{2} \alpha\right)}$. The minimum value is $\frac{8}{3} \pi R^{3}$, so the minimum value of $k$ is $\frac{4}{3}$.