Question

A cylindrical container of radius 6 cm and height 15 cm is filled with ice-cream. The whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height of the conical portion is 4 times the radius of its base, then find the radius of the ice-cream cone.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{container},R=6\mathrm{cm}, \\ \mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{container},H=15\mathrm{cm}, \\ \mathrm{Let}\mathrm{the}\mathrm{base}\mathrm{radius}\mathrm{and}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{ice}-\mathrm{cream}\mathrm{cone}\mathrm{be}r\mathrm{and}h,\mathrm{respectively}. \\ \mathrm{Also},h=4r \\ \mathrm{Now},\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{container}=\mathrm{\pi }{R}^{2}H \\ =\frac{22}{7}\times 6\times 6\times 15 \\ =\frac{11880}{7}{\mathrm{cm}}^3 \\ \Rightarrow \mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{ice}-\mathrm{cream}\mathrm{distributed}\mathrm{to}10\mathrm{children}=\frac{11880}{7}{\mathrm{cm}}^3 \\ \Rightarrow 10\times \mathrm{Volume}\mathrm{of}\mathrm{a}\mathrm{ice}-\mathrm{cream}\mathrm{cone}=\frac{11880}{7} \\ \Rightarrow 10\times \left(\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cone}+\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{hemisphere}\right)=\frac{11880}{7} \\ \Rightarrow 10\times \left(\frac{1}{3}\mathrm{\pi }{r}^{2}h+\frac{2}{3}\mathrm{\pi }{r}^3\right)=\frac{11880}{7} \\ \Rightarrow 10\times \left(\frac{1}{3}\mathrm{\pi }{r}^2\times 4r+\frac{2}{3}\mathrm{\pi }{r}^3\right)=\frac{11880}{7} \\ \Rightarrow 10\times \left(\frac{4}{3}\mathrm{\pi }{r}^3+\frac{2}{3}\mathrm{\pi }{r}^3\right)=\frac{11880}{7} \\ \Rightarrow 10\times \left(\frac{6}{3}\mathrm{\pi }{r}^3\right)=\frac{11880}{7} \\ \Rightarrow 10\times 2\times \frac{22}{7}\times r^3=\frac{11880}{7} \\ \Rightarrow r^3=\frac{11880\times 7}{7\times 10\times 2\times 22} \\ \Rightarrow r^3=27 \\ \Rightarrow r=\sqrt[3]{27} \\ \therefore r=3\mathrm{cm} \end{gathered}$$

So, the radius of the ice-cream cone is 3 cm.