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 $14cm$