a cylinder and cone have the same radius if the ratio of the total surface area of cylinder and cone is 2:1 what is the ratio of height of cylinder and slant height of triangle

Dear student,
According to the question,
$\frac{2\pi r(h+r)}{\pi rl+\pi r^2}=\frac{2}{1}$
where r=radius  of cylinder and cone
l= slant height of cone
Therefore,

$$\begin{gathered} \frac{2\pi r(h+r)}{\pi rl+\pi r^2}=\frac{2}{1} \\ \frac{2\pi r(h+r)}{\pi r(l+r)}=\frac{2}{1} \\ \frac{h+r}{l+r}=\frac{1}{1}=1 \\ h+r=r+l \\ h=l \end{gathered}$$

Now,
​​​​​​Ratio of height of cylinder and slant height of triangle:
$$\begin{gathered} \frac{h}{l}=\frac{l}{l} \\ h:l=1:1 \end{gathered}$$
Regards