As per the question given,
Internal diameter of the hollow sphere $=6cm$
The internal radius of the hollow sphere $=6/2cm=3cm=r$
External diameter of the hollow sphere $=10cm$
Then, the external radius of the hollow sphere $=10/2cm=5cm=R$
As we know that,
Volume of the hollow spherical shell $=4/3\pi \times \left( {{R}^{3}}-{{r}^{3}} \right)$
$=4/3\pi \times \left( {{5}^{3}}-{{3}^{3}} \right)$ ….. (i)
It is given, the length of the solid cylinder $=8/3cm$
Let the radius of the solid cylinder be r cm
As we know that,
Formula for volume of the cylinder $=\pi \times {{r}^{2}}\times h$
$=\pi \times {{r}^{2}}\times 8/3$….. (ii)
Now equating both (i) and (ii), we have
$4/3\pi \times {{5}^{3}}-{{3}^{3}}=\pi \times {{r}^{2}}\times 8/3$
$4/3\times \left( 125-27 \right)={{r}^{2}}\times 8/3$
$98/2={{r}^{2}}$
${{r}^{2}}=49$
$r=7$
So, $d=7\times 2=14cm$
Therefore, the diameter of the cylinder is $14cm$