Question

A hollow sphere of internal and external diameters $4$ cm and $8$ cm respectively, is melted into a cone of base diameter $8$ cm. The height of the cone is:

• A. $4$ cm
• B. $14$ cm
• C. $16$ cm
• D. $56$ cm

Answer 1

The correct option is B $14$ cm

External radius of the hollow sphere $=\frac{8}{2}=4$ cm

Internal radius of the hollow sphere $=\frac{4}{2}=2$ cm

$\therefore$ Volume of the metal $=$ Volume of external sphere $-$ Volume of internal sphere

$=\frac{4}{3}\pi {\left(4\right)}^3-\frac{4}{3}\pi {\left(2\right)}^3$

$=\frac{4}{3}\pi [4^3-2^3]$

$=\frac{4}{3}\pi [64-8]$

$=\frac{4}{3}\pi \left(56\right){cm}^3$

Radius of the cone $=\frac{8}{2}$ cm $=4$ cm

Let $h$ be the height of the cone.

Then volume of the cone $=\frac{1}{3}\pi {\left(4\right)}^{2}h{cm}^3$

Since the metal of the spherical shell is to be converted into the conical solid

$\therefore \frac{4}{3}\pi \left(56\right)=\frac{1}{3}\pi \left(16\right)h\Rightarrow h=\frac{4\times 56}{16}=14$ cm.

Hence, the height of the cone is $14$ cm.