A solid cylinder has a total surface area of $231{\text{ m}}^2$ . Its curved surface area is

$\frac{2}{3}$ of the total surface area. The volume of the cylinder is

For solid cylinder,

T.S.A $=231{\text{ m}}^2$

$2\pi r\left(r+h\right)=231{\text{m}}^2\to \left(1\right)$

C.S.A $=\frac{2}{3}$ T.S.A

C.S.A$=\frac{2}{3}\times 231$

$2\pi rh=2\times 77$

$2\pi rh=154\to \left(2\right)$

Substituting equation (2) in equation (1), we get

$2\pi r^2+2\pi rh=231$

$2\pi r^2+154=231$

$2\pi r^2=231-154$

$2\pi r^2=77$

$r^2=\frac{77}{2}\times \frac{7}{22}$

$r=\frac{7}{2}$ m

sub$r=\frac{7}{2}$ in equation (2), we get

$2\times \frac{22}{7}\times \frac{7}{2}\times h=154$

$h=\frac{154}{22}$

$h=7$ m

$\therefore$ volume of cylinder $=\pi r^{2}h$

$=\frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\times 7$

$=\frac{539}{2}{\text{ cm}}^3$

$=$ $269\frac{1}{2}{\text{ m}}^3$