2013-I-13

1. Since the base area of a larger circular cylinder is $9$ times that of a smaller one, we have

   $\begin{array}{rcl}
   \left(\dfrac{r}{R}\right)^2 & = & \dfrac{1}{9} \\
   r:R & = & 1:3
   \end{array}$

2. By the result of (a)(i), we have $R=3r$. Let $H\text{ cm}$ be the height of a larger circular cylinder.

   $\begin{array}{rcl}
   2\times \pi R^2 \times H & = & 27 \times \pi r^2 \times 10 \\
   2\times (3r)^2 \times H & = & 270 r^2 \\
   H & = & 15
   \end{array}$

   Therefore, the height of a larger circular cylinder is $15\text{ cm}$.

$\begin{array}{cl}
& \dfrac{\text{the base radius of smaller cylinder}}{\text{the base radius of larger cylinder}} \\
= & \dfrac{r}{R} \\
= & \dfrac{1}{3}
\end{array}$

And by the result of (a)(ii),

$\begin{array}{cl}
& \dfrac{\text{the height of smaller cylinder}}{\text{the height of larger cylinder}} \\
= & \dfrac{10}{15} \\
= & \dfrac{2}{3}
\end{array}$

Since the two ratio above are not equal, a smaller circular cylinder and a larger circular cylinder are not similar. Therefore, I don’t agree.