Ex.13.2 Q3 Surface Areas and Volumes Solution - NCERT Maths Class 10

Go back to  'Ex.13.2'

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 \(5\,\rm{cm}\) and diameter \(2.8\,\rm{cm}\) (see Fig. 13.15).

Text Solution

 

What is known?

Each Gulab jamun’s shape is like a cylinder with \(2\) hemispherical ends with length \(5\rm{cm}\) and diameter \(2.8\rm{cm.}\)
Each Gulab jamun contains sugar syrup up to about \(30\%\) of its volume.

What is unknown?

Volume of sugar syrup in \(45\) Gulab jamuns (approximately)

Reasoning:

Draw the figure to visualize the shape of Gulab jamun

From the figure it’s clear that

Length of cylindrical part \(=\) length of a Gulab jamun \( - {\rm{ }}2 \times \) radius of the hemispherical part

Also,

Diameter of Gulab jamun \(=\) diameter of cylindrical part

Radius of cylindrical part \(=\) radius of hemispherical part

 In order to find volume of sugar syrup in \(45\) Gulab jamun we find volume of Gulab jamun

 Using the statement

The volume of the solid formed by joining 2 basic solids will actually be sum of the volumes of its constituents.

 Volume of \(1\) Gulab jamun \(=\) volume of cylindrical part \(+\) volume of the \(2\) hemispherical parts

 Since, it’s given a Gulab jamun contains sugar syrup up to about \(30\%\) of its volume. We make an assumption to every Gulab jamun contains sugar syrup at \(30\%\) of its volume to simplify calculation

 Volume of sugar syrup \(= 30\%\) of volume of \(45\) Gulab jamun.

Steps:

Diameter of the Gulab jamun,\(d = 2.8cm\)

Radius of cylindrical part \(=\) radius of hemispherical part\(\begin{align} = r = \frac{{2.8cm}}{2} = 1.4cm\end{align}\)

Length of cylindrical part,\(h = 5cm - 2 \times 1.4cm = 2.2cm\)

Volume of \(1\) Gulab jamun \(=\) volume of cylindrical part\( + {\rm{ 2}} \times \) volume of the hemispherical parts

\[\begin{align}&= \pi {r^2}h + 2 \times \frac{2}{3}\pi {r^3}\\&= \pi {r^2}h + \frac{4}{3}\pi {r^3}\\&= \pi {r^2}\left( {h + \frac{4}{3}r} \right)\\&= \frac{{22}}{7} \times 1.4cm \times 1.4cm \times \left( {2.2cm + \frac{4}{3} \times 1.4cm} \right)\\&= \frac{{22}}{7} \times 1.4cm \times 1.4cm \times \frac{{122.2}}{3}cm\\&= \frac{{75.152}}{3}c{m^3}\end{align}\]

Volume of \(45\) Gulab jamuns \( = 45 \times \) volume of \(1\) Gulab jamun

\[\begin{align}&= 45 \times \frac{{75.152}}{3}c{m^3}\\&= 15 \times 75.152c{m^3}\\&= 1127.28c{m^3}\end{align}\]

Volume of sugar syrup in \(45\) Gulab jamuns\( = 30\% \)   of volume of \(45\) Gulab jamun

\[\begin{align}&= \frac{{30}}{{100}} \times 1127.28c{m^{^3}}\\ &= 338.184c{m^3}\\ &= 338{\rm{ }}c{m^3}({\text{approximately}})\end{align}\]