Question

A sphere is placed in a cone of height $9$cm as shown in the figure. The volume of the sphere, if it touches lateral surface and base of the cone, is

• A. $3\pi c{m}^3$
• B. $12\pi c{m}^3$
• C. $24\pi c{m}^3$
• D. $36\pi c{m}^3$

Answer 1

Answer: (D)

$36\pi c{m}^3$

In, $△OCD=,\angle COD=30°$

$\therefore \sin 30°=\frac{CD}{DO}\Rightarrow \frac{1}{2}=\frac{CD}{9}\Rightarrow CD=4.5cm$

We know that $△OAB\sim △OCD$

$\frac{OA}{OC}=\frac{AB}{CD}\Rightarrow \frac{9-AB}{9}=\frac{AB}{4.5}[\because OC=OA+AC=OA+AB]$

$\Rightarrow 40.5-4.5AB=9AB\Rightarrow 13.5AB=40.5\Rightarrow AB=\frac{40.5}{13.5}=3cm$

Volume of sphere$=\frac{4}{3}\pi r^3=\frac{4}{3}\times \pi \times 3\times 3\times 3=36\pi c{m}^3$