A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see figure).
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Given:

A gulab jamun contains sugar syrup up to about 30% of its volume.

To do:

We have to find how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with a length 5 cm and a diameter of 2.8 cm

Solution:

The volume of one piece of gulab jamun

$=$ Volume of the cylindrical portion $+$ Volume of two hemispherical ends

Radius of each hemispherical portion $= \frac{2.8}{2}$

$= 1.4\ cm$

Volume of one hemispherical end $=\frac{2}{3} \pi r^{3}$

$=\frac{2}{3} \times \frac{22}{7}(1.4)^{3}$

$=\frac{2}{3} \times \frac{22}{7} \times (1.4)^3$

$=\frac{2 \times 22 \times 2 \times 14 \times 14}{3 \times 10 \times 10 \times 10}$

$=5.74 \mathrm{~cm}^{3}$

Volume of both hemispherical ends $= 2 \times 5.74$

$= 11.48\ cm^3$

Height of the cylindrical portion $=$ Total height $-$ Radius of both hemispherical ends

$= 5-2(1.4)\ cm$

$= 5-2.8$

$= 2.2\ cm$

The radius of the cylindrical portion $= 1.4\ cm$

The volume of the cylindrical portion of gulab jamun $= \pi r^2h$

$= \frac{22}{7} \times (1.4)^2 \times 2.2$

$= 22\times2\times1.4\times2.2$

$= 13.55\ cm^3$

The total volume of one gulab jamun $=$ Volume of the two hemispherical ends $+$ Volume of the cylindrical portion

$= 11.48+ 13.55$

$= 25.03\ cm^3$

The volume of sugar syrup $= 30 \%$ of the volume of gulab jamun

$= \frac{30}{100} \times 25.03$

$= 7.50\ cm^3$

Therefore,

The volume of sugar syrup in 45 gulab jamuns

$= 45 \times$ (volume of sugar syrup in one gulab jamun)

$= 45 \times 7.50$

$= 337.5\ cm^3$

$= 338\ cm^3$

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Updated on: 10-Oct-2022

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