Solution:
The ratio of the radius to the height of the cylinder is

Let the radius be

Let the height be

The volume of the cylinder is given below as

Concept:
The volume of a cylinder is given below as

By substituting values, we will have
![\begin{gathered} V_{\text{cylinder}}=\pi* r^2* h \\ 1617=(22)/(7)*(2x)^2*(3x) \\ 1617=(22)/(7)*4x^2*3x \\ 1617*7=264x^3 \\ \text{divdie both sides by 264} \\ (264x^3)/(264)=(1617*7)/(264) \\ x^3=(343)/(8) \\ x=\sqrt[3]{(343)/(8)} \\ x=(7)/(2) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/tg1lk4eamsnz1flni2w0ts8dlgaoip3nqx.png)
The radius therefore will be

The height of the cylinder will be

The formula for the total surface area of a cylinder is given below as

By substituting the values, we will have

Hence,
The total surface area of the cylinder is = 770cm²