The height of cone is 30cm A small cone is cut off at the topby a plane parallel to - Maths - Surface Areas and Volumes

Question

The height of cone is 30cm A small cone is cut off at the topby
a plane parallel to the base If its volume is {1/ 27}th of the
volume of given cone At what height above the base is the section
made

Ajanta Trivedi · Accepted Answer

the height of the cone ABC be H = 30 cm let the radius of the cone ABC be R volume of the bigger cone ABC = $\frac{1}{3} π R^{2} H = \frac{1}{3} π R^{2} * 30 = 10 π R^{2}$ (1) let the another cone is cut off at A'B' parallel to AB clearly triangle CO'B' and COB are similar triangles [since O'B' is parallel to OB] therefore $\frac{A O'}{O' B'} = \frac{A O}{O B} \Rightarrow \frac{h}{r} = \frac{30}{R} \Rightarrow \frac{r}{R} = \frac{h}{30} (2)$ let the height and radius of the smaller cone be h and r respectively the volume of the smaller cone is $\frac{1}{3} π r^{2} h$ $\frac{v o l u m e o f t h e s m a l l e r c o n e}{v o l u m e o f t h e b i g g e r c o n e} = \frac{1 / 3 π r^{2} h}{10 π R^{2}} = \frac{1}{27} \frac{r^{2} h}{30 R^{2}} = \frac{1}{27} (\frac{r}{R})^{2} h = \frac{30}{27} = \frac{10}{9} (3)$ using result (2) $(\frac{h}{30})^{2} h = \frac{10}{9} \frac{h^{3}}{900} = \frac{10}{9} h^{3} = \frac{10}{9} * 900 = 1000 h = \sqrt[3]{1000} = 10 c m$ therefore the height of the smaller cone is 10 cm i e at the height of 30-10=20 cm above the base section is made

The height of cone is 30cm. A small cone is cut off at the topby a plane parallel to the base. If its volume is {1/ 27}th of the volume of given cone. At what height above the base is the section made