Question

A container shaped like a right circular cylinder having diameter $12cm$ and height $15cm$ is full of ice cream. The ice cream is to be filled into cones of height $12cm$ and diameter $6cm$, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Answer 1

Step 1 - Finding the volume of the cylindrical container

The radius of the container = $\frac{d}{2}=\frac{12}{2}=6cm$

$$\begin{gathered} Volumeofthecylindricalcontainer={\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{container}=\frac{22}{7}\times 6^2\times 15 \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{container}=\frac{11880}{7}{\mathrm{cm}}^2 \end{gathered}$$

Step 2- Finding the volume of the complete shape

Radius of the shape = $\frac{d}{2}=\frac{6}{2}=3cm$

Height of cone shape = $12cm$

$$\begin{gathered} Volumeoftheice-creamshape=Volumeofcone+Volumeofhemisphere \\ Volumeoftheice-creamshape=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+\frac{2}{3}{\mathrm{\pi r}}^3 \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{ice}-\mathrm{cream}\mathrm{shape}=\frac{1}{3}{\mathrm{\pi r}}^2\left(\mathrm{h}+2\mathrm{r}\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{ice}-\mathrm{cream}\mathrm{shape}=\frac{1}{3}\times \frac{22}{7}\times 3\times 3\times \left(12+2\times 3\right) \\ \mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{ice}-\mathrm{cream}\mathrm{shape}=\frac{1188}{7}{\mathrm{cm}}^3 \end{gathered}$$

Step 3- Number of cones that can be filled from the cylindrical container

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{ice}-\mathrm{cream}\mathrm{cones}=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cylindrical}\mathrm{container}}{\mathrm{Volume}\mathrm{of}\mathrm{one}\mathrm{cone}} \\ \mathrm{Number}\mathrm{of}\mathrm{ice}-\mathrm{cream}\mathrm{cones}=\frac{{\displaystyle \frac{11880}{7}}}{{\displaystyle \frac{1188}{7}}}=10 \end{gathered}$$

Hence, the number of ice-cream cones that can be filled with the cylindrical container is $10$.