The internal and external diameters of a hollow hemispherical vessel are $ 21 \mathrm{~cm} $ and $ 25.2 \mathrm{~cm} $ respectively. The cost of painting $ 1 \mathrm{~cm}^{2} $ of the surface is 10 paise. Find the total cost to paint the vessel all over.

Given:

The internal and external diameters of a hollow hemispherical vessel are \( 21 \mathrm{~cm} \) and \( 25.2 \mathrm{~cm} \) respectively.

The cost of painting \( 1 \mathrm{~cm}^{2} \) of the surface is 10 paise.

To do:

We have to find the total cost to paint the vessel all over.

Solution:

External diameter of the hollow hemispherical vessel $= 25.2\ cm$

Internal diameter of the hollow hemispherical vessel $= 21\ cm$

This implies,

Outer radius $\mathrm{R}=\frac{25.2}{2}$

$=12.6 \mathrm{~cm}$

Inner radius $r=\frac{21}{2}$

$=10.5 \mathrm{~cm}$

Total surface area of the vessel $=$ Outer surface area $+$ Inner surface area $+$ Area of the base

$=2 \pi \mathrm{R}^{2}+2 \pi r^{2}+(\pi \mathrm{R}^{2}-\pi r^{2})$

$=2 \pi \mathrm{R}^{2}+2 \pi r^{2}+\pi \mathrm{R}^{2}-\pi r^{2}$

$=3 \pi \mathrm{R}^{2}+\pi r^{2}$

$=3 \times \frac{22}{7}(12.6)^{2}+\frac{22}{7}(10.5)^{2}$

$=\frac{66}{7} \times 158.76+\frac{22}{7} \times 110.25$

$=1496.88+346.50 \mathrm{~cm}^{2}$

$=1843.38 \mathrm{~cm}^{2}$

Cost of painting the surface per $1\ cm^2=10$ paise

$=Rs.\ 0.10$

Total cost to paint the vessel all over $=Rs.\ 1843.38\times0.10$

$=Rs.\ 184.338$

$=Rs.\ 184.34$

The total cost to paint the vessel all over is Rs. 184.34.

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Updated on: 10-Oct-2022

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