Question

A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter $4\frac{2}{3}$ cm and height 3 cm. Find the number of cones so formed.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{metallic}\mathrm{sphere},R=\frac{28}{2}=14\mathrm{cm}, \\ \mathrm{Radius}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone},r=\frac{1}{2}\times \left(4\frac{2}{3}\right)=\frac{1}{2}\times \frac{14}{3}=\frac{7}{3}\mathrm{cm}\mathrm{and} \\ \mathrm{Height}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone},h=3\mathrm{cm} \\ \mathrm{Now}, \\ \mathrm{The}\mathrm{number}\mathrm{of}\mathrm{cones}\mathrm{so}\mathrm{formed}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{metallic}\mathrm{sphere}}{\mathrm{Volume}\mathrm{of}\mathrm{a}\mathrm{smaller}\mathrm{cone}} \\ =\frac{\left({\displaystyle \frac{4}{3}}\mathrm{\pi }{R}^3\right)}{\left({\displaystyle \frac{1}{3}}\mathrm{\pi }{r}^{2}h\right)} \\ =\frac{{\displaystyle 4R^3}}{{\displaystyle r^{2}h}} \\ =\frac{4\times 14\times 14\times 14}{\left({\displaystyle \frac{7}{3}}\times {\displaystyle \frac{7}{3}}\times 3\right)} \\ =672 \end{gathered}$$

So, the number of cones so formed is 672.

Disclaimer: The answer given in the textbook is incorrect. The same has been corrected above.