11. (12 points) There is a cylinder, the height of which is 3 times the radius of the base. It is divided into two cylinders, a larger one and a smaller one, as shown in the figure. The surface area of the larger cylinder is 3 times that of the smaller cylinder. Therefore, the volume of the larger cylinder is $\qquad$ times that of the smaller cylinder.
【Solution】Solution: Let the radius of the base of this cylinder be $r$, then the height is $3 r$, the height of the small cylinder is $h$, so the height of the large cylinder is $(3 r-h)$;

Since the surface area of the large cylinder is 3 times that of the small cylinder,
thus $h=\frac{\mathrm{r}}{4}$, then the height of the large cylinder is $\frac{11}{4} r$;
and since the base areas of the two cylinders are the same,
$$
\frac{11}{4} r \div \frac{r}{4}=11 \text {, }
$$

so the volume of the large cylinder is also 11 times that of the small cylinder.
Therefore, the answer is: 11.