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.

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Solution

## Step-1: Solve for the number of cones:Given: Radius of cylinder $\left(R\right)=\frac{12}{2}⇒6cm$height of cylinder $\left(H\right)=15cm$radius of cone $\left(r\right)=\frac{6}{2}cm⇒3cm$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 equationvolume of ice cream cones$=volumeofcone+volumeofhemisphere$ $=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}+\frac{2}{3}{\mathrm{\pi r}}^{3}$ $=\frac{1}{3}{\mathrm{\pi r}}^{2}\left(\mathrm{h}+2\mathrm{r}\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{3}\mathrm{\pi }{\left(3\right)}^{2}\left(12+2×3\right)\phantom{\rule{0ex}{0ex}}=\mathrm{\pi }×3×18\phantom{\rule{0ex}{0ex}}=54\mathrm{\pi }$Number of ice cream cones$=\frac{volumeofcylinder}{volumeoficecreamcones}$ $=\frac{\mathrm{\pi }×{\left(6\right)}^{2}×15}{54\mathrm{\pi }}\phantom{\rule{0ex}{0ex}}=\frac{36×15}{54}\phantom{\rule{0ex}{0ex}}=10$Hence, number of cones are $10$.

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