A wooden bookshelf has external dimensions as follows: Height = 10 cm, Depth = 25cm, Breadth = 85 cm (See in Figure). The thickness of the plank is 5 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 `c m^2` and the rate of painting is 10 paise per `c m^2` . Find the total expenses required for polishing and painting the surface of the bookshelf.

Explanation

Since the bookshelf is cuboidal in shape and opened at the front with three shelves of the same dimensions, the area to be polished will be 5 surfaces of cuboidal bookshelf and the front border with plank’s thickness.
The area to be painted will be 3 shelves of the bookshelf with internal dimensions and the area of each shelf will be 5 surfaces of the cuboidal shelf.
We can calculate the total cost of polishing and painting by multiplying the rate and their respective area.
The Volume of a cuboid of length l, breadth b, and height h, is = `l xx b xx h` External measures of the bookshelf is,
Breadth, B = `85` cm
Depth, D = 25 cm
Height, `H = 110 cm`
The thickness of the plank, `t = 5 cm`
Internal measures of the bookshelf is,
The breadth of each shelf, b = B - 2t
`b = 85 cm - 2 xx 5 cm = 75 cm`
Depth of each shelf, `d = D - t`
` d = 25 cm - 5 cm = 20 cm`
Height of each shelf, `h = H - 4t`
`= (110 cm - 4 xx 5 cm) div 3`

`= 90/3 cm = 30 cm`

Now, Surface area to be polished = External 5 surfaces of the bookshelf + border of the shelf
`= 2(B + H) D + BH + 2Ht + 4bt`

`= [2 xx (85 cm + 110 cm) xx 25 cm] + (85 cm xx 110 cm) + (2 xx 110 cm xx 5 cm) + (4 xx 75 cm xx 5 cm)`
`= 9750 cm^2 + 9350 cm^2 + 1100 cm^2 + 1500 cm^2`

`= 21700 cm^2`
Cost of polishing at the rate of 20 paise per `cm^2 = 21700 cm^2 xx (₹ 20/100) / cm^2 = ₹ 4340`
Surface area to be painted = Internal 5 surfaces of 3 shelves
= 3 [2(b + h)d + bh]

`= 3 [2 xx (75 cm + 30 cm) xx 20 cm + (75 cm xx 30 cm)]`

`= 3 [4200cm^2 + 2250 cm^2]`
`= 3 xx 6450 cm^2`

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