A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is $ 104 \mathrm{~cm} $ and the radius of each of the hemispherical ends is $ 7 \mathrm{~cm} $, find the cost of polishing its surface at the rate of $ ₹ 10 $ per $ \mathrm{dm}^{2} $.

Given:

A solid is composed of a cylinder with hemispherical ends.

The whole length of the solid is \( 104 \mathrm{~cm} \) and the radius of each of the hemispherical ends is \( 7 \mathrm{~cm} \).

To do:

We have to find the cost of polishing its surface at the rate of \( ₹ 10 \) per \( \mathrm{dm}^{2} \).

Solution:

Total height(length) of the solid $= 104\ cm$

Radius of each hemispherical end $r = 7\ cm$
Height of the cylindrical part $h= 104 - 2 \times 7\ cm$

$= 104- 14\ cm$

$= 90\ cm$
Total outer surface area $=$ Curved surface area of the cylindrical part $+ 2 \times$ Curved surface area of each hemispherical part

$=2 \pi r h+2 \times 2 \pi r^{2}$

$=2 \pi r h+4 \pi r^{2}$

$=2 \pi r(h+2 r)$

$=2 \times \frac{22}{7} \times 7(90+2 \times 7)$

$=44\times104$

$=4576 \mathrm{~cm}^{2}$

$=\frac{4576}{100} \mathrm{dm}^{2}$

$=45.76 \mathrm{dm}^{2}$

Cost of polishing the surface per $\mathrm{dm}^{2} = Rs.\ 10$

Therefore,

Total cost of polishing $= Rs.\ 45.76 \times 10$

$= Rs.\ 457.60$

The cost of polishing its surface at the rate of \( ₹ 10 \) per \( \mathrm{dm}^{2} \) is Rs. 457.60.

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Updated on: 10-Oct-2022

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