Ex.13.1 Q6 Surface Areas and Volumes Solution - NCERT Maths Class 10

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Question

A medicine capsule is in the shape of cylinder with two hemispheres stuck to each of its ends (see Fig. 13.10). The length of the entire capsule is \(14 \,\rm{mm}\) and the diameter of the capsule is \(5 \,\rm{mm}\). Find its surface area.

Text Solution

What is known?

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to its ends. The length of the entire capsule \(=14 \,\rm{mm}\), diameter of the capsule \(=5 \,\rm{mm}\).

What is the unknown?

The surface area of the capsule.

Reasoning:

Since the capsule is in shape of a cylinder with \(2\) hemispheres stuck to its ends,

 Diameter of the capsule \(=\) diameter of its cylindrical part \(=\) diameter of its hemispherical part.

From the figure, it’s clear that the capsule has the curved surface of two hemispheres and the curved surface of a cylinder.

Surface area of the capsule \( = 2 \times\) CSA of hemispherical part \(+\) CSA of cylindrical part

We will find the surface area of the capsule by using formulae;
CSA of the hemisphere  \( = 2\pi {r^2}\)
where \(r\) is the radius of the hemisphere
CSA of the cylinder \( = 2\pi rh\) 
where \(r\) and \(h\) are radius and height of the cylinder respectively.

Length of the cylindrical part \(=\) Length of the capsule \(- 2 \times \) radius of the hemispherical part

Steps:

Diameter of the capsule,  \(d = 5mm\)
Radius of the hemisphere, \(\begin{align}r = \frac{d}{2} = \frac{5}{2}mm\end{align}\) 
Radius of the cylinder,  \(\begin{align}r = \frac{5}{2}mm\end{align}\)
Length of the cylinder = Length of the capsule\(- 2 \times \) radius of the hemisphere

\[h = 14mm - 2 \times \frac{5}{2}mm = 9mm\]

Surface area of the capsule\(- 2 \times \) CSA of hemispherical part \(+\) CSA of cylindrical part

\[\begin{align}& = 2 \times 2\pi {r^2} + 2\pi rh\\ &= 2\pi r\left( {2r + h} \right)\\ &= 2 \times \frac{{22}}{7} \times \frac{5}{2}{mm} \times \left( {2 \times \frac{5}{2}{mm} + 9{mm}} \right)\\&= \frac{{110}}{7}{mm} \times 14{mm}\\&= 220{m{m^2}}\end{align}\]