Suppose that the original can has radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$. Since the surface area of the original can is $300 \mathrm{~cm}^{2}$, then $2 \pi r^{2}+2 \pi r h=300$. When the radius of the original can is doubled, its new radius is $2 r \mathrm{~cm}$, and so an expression for its surface area, in $\mathrm{cm}^{2}$, is $2 \pi(2 r)^{2}+2 \pi(2 r) h$ which equals $8 \pi r^{2}+4 \pi r h$, and so $8 \pi r^{2}+4 \pi r h=900$. When the height of the original can is doubled, its new height is $2 h \mathrm{~cm}$, and so an expression for its surface area, in $\mathrm{cm}^{2}$, is $2 \pi r^{2}+2 \pi r(2 h)$ which equals $2 \pi r^{2}+4 \pi r h$. Multiplying $2 \pi r^{2}+2 \pi r h=300$ by 3, we obtain $6 \pi r^{2}+6 \pi r h=900$. Since $8 \pi r^{2}+4 \pi r h=900$, we obtain $6 \pi r^{2}+6 \pi r h=8 \pi r^{2}+4 \pi r h$. Therefore, $2 \pi r h=2 \pi r^{2}$, and $\pi r h=\pi r^{2}$. Since $2 \pi r^{2}+2 \pi r h=300$ and $\pi r h=\pi r^{2}$, then $2 \pi r^{2}+2 \pi r^{2}=300$ and so $4 \pi r^{2}=300$ or $\pi r^{2}=75$. Since $\pi r h=\pi r^{2}=75$, then $2 \pi r^{2}+4 \pi r h=6 \cdot 75=450$, and so the surface area of the cylinder with its height doubled is $450 \mathrm{~cm}^{2}$.