The diameters of internal and external surfaces of a hollow spherical

Question:

The diameters of internal and external surfaces of a hollow spherical shell are $10 \mathrm{~cm}$ and $6 \mathrm{~cm}$ respectively. If it is melted and recast into a solid cylinder of length of $2 \frac{2}{3} \mathrm{~cm}$, find the diameter of the cylinder.

Solution:

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

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

$=\frac{4}{3} \pi \times 98 \mathrm{~cm}^{3}$

The spherical shell is melted to recast a solid cylinder of length $\frac{8}{3} \mathrm{~cm}$. Let the radius of the solid cylinder is $r \mathrm{~cm}$. Therefore, the volume of the solid cylinder is

$V_{1}=\pi \times(r)^{2} \times \frac{8}{3} \mathrm{~cm}^{3}$

Since, the volume of the hollow spherical shell is equal to the volume of the solid cylinder; we have

$V_{1}=V$

$\Rightarrow \pi \times(r)^{2} \times \frac{8}{3}=\frac{4}{3} \pi \times 98$

$\Rightarrow \quad r^{2}=49$

$\Rightarrow \quad r=7$

Hence, the diameter of the solid cylinder is two times its radius, which is 14 cm.

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