# Ex.13.2 Q8 Surface Areas and Volumes Solution - NCERT Maths Class 10

## Question

A spherical glass vessel has a cylindrical neck \(8\rm\,{ cm}\) long, \(2\rm\,{ cm}\) in diameter; the diameter of the spherical part is \(8.5\rm\,{ cm}\). By measuring the amount of water it holds, a child finds its volume to be \(345\rm\,{ cm^3}\). Check whether she is correct, taking the above as the inside measurements, and \(\pi = 3.14.\)

## Text Solution

**What is known?**

A spherical glass vessel with a cylindrical neck of length \(=8\rm\,{ cm}\) and diameter \(=2\rm\,{ cm}\)

The diameter of the spherical part \(=8.5\rm\,{ cm}\)

According to the child volume of the vessel \(345\rm{cm}^3\)

**What is unknown?**

Whether the child is correct in saying the volume of the glass vessel is \(345\,{\text{c}}{{\text{m}}^3}\)

**Reasoning:**

Draw a figure to visualize the glass vessel

Since the glass vessel is a combination of a sphere and a cylinder

Volume of glass vessel \(=\) Volume of the spherical part \(+\) Volume of the cylindrical part

**Steps:**

Height of cylindrical part,\(h = 8 \rm cm\)

Radius of cylindrical part,\(\begin{align}{r_1} = \frac{{2 \rm cm}}{2} = 1 \rm cm\end{align}\)

Radius of spherical part,\(\begin{align}{r_2} = \frac{{8.5 \rm cm}}{2} = 4.25 \rm cm\end{align}\)

Volume of the glass vessel \(=\) volume of the spherical part \(+\) volume of the cylindrical part

\[\begin{align}&= \frac{4}{3}\pi r_2^3 + \pi r_1^2h\\&= \pi \left( {\frac{4}{3}r_2^3 + r_1^2h} \right)\\

&= 3.14 \!\! \times \!\!\begin{pmatrix} \frac{4}{3} \times 4.25 \rm cm \\ \times 4.25 \rm cm \\ \times 4.25 \rm cm \\ + 1 \rm cm \\ \times 1 \rm cm \\ \times 8 \rm cm \end{pmatrix} \\&= \begin{Bmatrix} 3.14 \\\times \begin{pmatrix} 102.354 \rm c{m^3} \\ + 8 \rm c{m^3} \end{pmatrix} \end{Bmatrix} \\&= 3.14 \times 110.354 \rm c{m^3}\\&= 346.51 \rm c{m^3}\end{align}\]

Therefore, the child is not correct in saying that the volume of the glass vessel is \(345\rm{cm}^3\)