# The diameters of the internal and external surfaces

Question:

The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm. find the height of the cylinder.

Solution:

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 $\frac{8}{3} \mathrm{~cm}$