Question

A container shaped like a right circular cylinder having diameter $12$ cm and height $15$ cm is full of ice cream.

The ice cream is to be filled into cones of height $12$ cm and diameter $6$ cm, having a hemispherical shape on the top.

Find the number of such cones which can be filled with ice cream.

Answer 1

Step-1: Solve for the number of cones:

• Given: Radius of cylinder $\left(R\right)=\frac{12}{2}\Rightarrow 6cm$
• height of cylinder $\left(H\right)=15cm$
• radius of cone $\left(r\right)=\frac{6}{2}cm\Rightarrow 3cm$
• height of cone $\left(h\right)=12cm$

Let the total number of ice cream be $n$

Number of ice cream cones$=\frac{volumeofcylinder}{volumeoficecreamcones}$

Volume of cylinder$=\pi R^{2}H$

Step-2: Simplify the equation

volume of ice cream cones$=volumeofcone+volumeofhemisphere$

$=\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}+\frac{2}{3}{\mathrm{\pi r}}^3$

$$\begin{gathered} =\frac{1}{3}{\mathrm{\pi r}}^2(\mathrm{h}+2\mathrm{r}) \\ =\frac{1}{3}\mathrm{\pi }{\left(3\right)}^2\left(12+2\times 3\right) \\ =\mathrm{\pi }\times 3\times 18 \\ =54\mathrm{\pi } \end{gathered}$$

Number of ice cream cones$=\frac{volumeofcylinder}{volumeoficecreamcones}$

$$\begin{gathered} =\frac{\mathrm{\pi }\times {\left(6\right)}^2\times 15}{54\mathrm{\pi }} \\ =\frac{36\times 15}{54} \\ =10 \end{gathered}$$

Hence, number of cones are $10$.