A wooden bookshelf has external dimensions as follows: Height $ =110 \mathrm{~cm} $, Depth $ =25 \mathrm{~cm} $, Breadth $ =85 \mathrm{~cm} $ (see figure below). The thickness of the plank is $ 5 \mathrm{~cm} $ everywhere. The external faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per $ \mathrm{cm}^{2} $ and the rate of painting is 10 paise per $ \mathrm{cm}^{2} $, find the total expenses required for polishing and painting the surface of the bookshelf.
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Given:

A wooden bookshelf has external dimensions as follows:

Height $=110\ cm$

Depth $= 25\ cm$

Breadth $= 85\ cm$.

The thickness of the plank is $5\ cm$ everywhere.

The external faces are to be polished and the inner faces are to be painted.

The rate of polishing is $20$ paise per $cm^2$ and the rate of painting is $10$ paise per $cm^2$.

To do:

We have to find the total expenses required for polishing and painting the surface of the bookshelf.

Solution:

Thickness of the plank $= 5\ cm$

Surface area to be polished $= [(100 \times 85) + 2 (110 \times 25) + 2 (85 \times 25) + 2 (110 \times 5) + 4 (75 \times 5)]$

$= (9350 + 5500 + 4250 + 1100 + 1500)$

$= 21700\ cm^2$

Total cost $=\frac{21700 \times 20}{100}$

$=Rs.\ 4340$

Area of internal surface $=$ Area of five faces of 3 cuboids

$=$ Total surface area $-$ Area of 3 cuboids

$=3[2(75 \times 30+30 \times 20+75 \times 20)]-3 \times(75 \times 30)$

$=6(2250+600+1500)-6750$

$=19350 \mathrm{~cm}^{2}$

Therefore,

Cost of painting at the rate of 10 paise per $\mathrm{cm}^{2} =\frac{19350 \times 10}{100}$

$=Rs.\ 1935$

Hence total cost $=Rs.\ (4340+1935)$

$=Rs.\ 6275$

The total expenses required for polishing and painting the surface of the bookshelf is $Rs.\ 6275$.

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Updated on: 10-Oct-2022

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