Question

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.

Answer 1

Given:
Radius $\left(r\right)$ of cylindrical part $=$ Radius $\left(r\right)$ of hemispherical part = $\frac{2.8}{2}=1.4cm$

Length of each hemispherical part $=$ Radius of hemispherical part $=1.4cm$

Length $\left(h\right)$ of cylindrical part $=5-2\times$ length of hemispherical part

$\Rightarrow 5-$ $2\times 1.4=2.2cm$

Volume of one gulab jamun $=$ Volume of cylindrical part $+2\times$ Volume of hemispherical part

We know that, volume of a cylinder is $\pi r^{2}h$

and volume of a hemisphere is $\frac{2}{3}\pi r^3$

$\therefore$ Volume of one gulab jamun $=\pi r^{2}h+2\times \frac{2}{3}\pi r^3=\pi r^{2}h+\frac{4}{3}\pi r^3$

$\Rightarrow \pi \times (1.4{)}^2\times 2.2+\frac{4}{3}\pi (1.4{)}^3$

$\Rightarrow \frac{22}{7}\times 1.4\times 1.4\times 2.2+\frac{4}{3}\times \frac{22}{7}\times 1.4\times 1.4\times 1.4[\because \pi =\frac{22}{7}]$

$\Rightarrow 13.552+11.498=25.05c{m}^3$

Hence, volume of one gulab jamun $=25.05c{m}^3$

So, volume of $45$ gulab jamuns $=45\times 25.05=1,127.25c{m}^3$

Also, given that, volume of sugar syrup $=30\%$ of volume of gulab jamun

$\therefore$ Volume of the sugar syrup of $45$ gulab jamuns $=\frac{30}{100}\times 1,127.25$

$\Rightarrow 338.17c{m}^3\approx 338c{m}^3$

Hence, the amount of syrup in $45$ gulab jamuns is approximately $338c{m}^3$.