The ratio between the radius of the base and height of the cylinder is $2:3$. If the volume is $12936 cm^{3}$, what is the total surface area of the cylinder?

Let the radius be $2x$ and the height be $3x$. The volume of a cylinder is $\pi r^2h$, so we have $(22/7) \times (2x)^2 \times (3x) = 12936$. This simplifies to $264x^3/7 = 12936$, which means $x^3 = 343$ and $x = 7$. The radius is $14$ cm and the height is $21$ cm. The total surface area formula is $2\pi r(r + h)$, which gives $2 \times (22/7) \times 14 \times (14 + 21) = 3080$ cm$^2$. The total surface area is 3080 cm$^2$.

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