Question

A cone of radius k and height h is divided into two parts by drawing a plane through one-third of its axis from base, parallel to the base, then the volume of new cone is

• A. $\frac{1}{9}\pi k^{2}h$
  
• B. $\frac{1}{27}\pi k^{2}h$
  
• C. $\frac{8}{81}\pi k^{2}h$
  
• D. $\frac{1}{243}\pi k^{2}h$
  

Answer 1

The correct option is C $\frac{8}{81}\pi k^{2}h$


Let ABC be the given cone. Let DE be the plane that divides the axis AO in the ratio 2 : 1.

∴ DE || BC and AF : FO = 2 : 1.

Let the height and the radius of new cone formed be hʹ and r respectively.

AF : FO = 2 : 1

⇒ AF : AO = 2 : 3

$\Rightarrow \frac{h^{\prime }}{h}=\frac{2}{3}$

$\Rightarrow h^{\prime }=\frac{2}{3}h$


Also, $\Delta AFE-\Delta AOC$

(If a line joining two sides of a triangle is parallel to the third side then, the triangles so formed are similar.)

$\Rightarrow \frac{AF}{AO}=\frac{FE}{OC}$ (Ratio of corresponding sides of two similar triangles are equal)

$\Rightarrow \frac{\frac{2h}{3}}{h}=\frac{r}{k}$

$\Rightarrow r=\frac{2k}{3}$

$\therefore$ Volume of new cone $=\frac{1}{3}\pi r^2{h}^{\prime }$

$=\frac{1}{3}\times \pi \times {\left(\frac{2k}{3}\right)}^2\times \frac{2h}{3}$

$=\frac{8}{81}\pi k^{2}h$

Hence, the correct answer is option (c).