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Surface Areas And Volumes — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsSurface Areas And Volumes3 Marks
Question
A cylindrical container is filled with ice-cream, whose diameter is 12cm and height is 15cm. the whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream.
✓

Answer

Volume of cylindrical container
$=\pi\text{r}^2\text{h}$
$=\pi\times(6)^2\times15$
Amount of ice-cream distributed to 10 children $=\frac{\pi\times(6)^2\times15}{10}$
Therefore,
Height of conical portion = 2 × diameter of its bars
Let the diameter of bare = r
Height = 2r
Therefore,
Volume of the cones
$=\frac{1}{3}\pi\Big(\frac{\text{r}}{2}\Big)^2\text{h}+\frac{2}{3000}\pi\text{r}^3$
$=\frac{1}{3}\pi\Big(\frac{\text{r}}{2}\Big)^2(\text{h}+2\text{r})$
$=\frac{1}{3}\pi\Big(\frac{\text{r}}{2}\Big)^2\Big(2\text{r}+2\times\frac{\text{r}}{2}\Big)$
$=\frac{1}{3}\pi\Big(\frac{\text{r}}{2}\Big)^2\times3$
$\text{r}=\frac{\pi\text{r}^3}{4}$
Therefore,
Volume of the cones = amount distributed
$\frac{\pi\text{r}^3}{4}=\frac{\pi(6)^2\times15}{10}$
$\text{r}^3=\frac{4\times6\times6\times15}{10}=4\times6\times9$
$\text{r}=\sqrt[3]{6\times6\times6}=6\text{cm}$

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