The diameters of the internal and external surfaces

The internal and external radii of the hollow spherical shell are 3cm and 5cm respectively. Therefore, the volume of the hollow spherical shell is

$V=\frac{4}{3} \pi \times\left\{(5)^{3}-(3)^{3}\right\} \mathrm{cm}^{3}$

The hollow spherical shell is melted to recast a cylinder of radius 7cm. Let, the height of the solid cylinder is h. Therefore, the volume of the solid cylinder is

$V_{1}=\pi \times(7)^{2} \times h \mathrm{~cm}^{3}$

Since, the volume of the solid cylinder is same as the volume of the hollow spherical shell, we have

$V_{1}=V$

$\Rightarrow \pi \times(7)^{2} \times h=\frac{4}{3} \pi \times\left\{(5)^{3}-(3)^{3}\right\}$

$\Rightarrow \quad 49 \times h=\frac{4}{3} \times 98$

$\Rightarrow \quad h=\frac{4 \times 98}{3 \times 49}$

$\Rightarrow \quad=\frac{8}{3}$

Therefore, the height of the solid cylinder is $\frac{8}{3} \mathrm{~cm}$