Question

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

• A. 14cm
• B. 10cm
• C. 6cm
• D. 4cm

Answer 1

Answer: (A)

14cm

Internal diameter of hollow sphere $=d=4cm$

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

External diameter of hollow sphere $=D=8cm$

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

Volume of the hollow sphere $=\frac{4}{3}\pi (R^3-r^3)$

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

$=\frac{4}{3}\pi (64-8)$

$=\frac{4}{3}\pi \times 56c{m}^3$

Diameter of the cone $=d_1=8cm$

Radius of the cone $=r_1=\frac{8}{2}=4cm$

Let the height of the cone be $h$ cm.

Volume of the cone $=\frac{1}{3}\pi r_1^{2}h$

$=\frac{1}{3}\pi \times 4^2\times h$

$=\frac{16\pi h}{3}$

Now,

Volume of the cone $=$ Volume of the hollow sphere

$\Rightarrow$ $\frac{16\pi h}{3}=\frac{4}{3}\pi \times 56$

$\Rightarrow$ $16h=4\times 56$

$\Rightarrow$ $h=\frac{4\times 56}{16}$

$\therefore$ $h=14cm$