Q1. A wooden bookshelf has external dimensions as follows: Height = 110cm, Depth = 25cm,

Breadth = 85cm (see fig. 13.31). The thickness of the plank is 5cm 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 cm² and the rate of painting is 10 paise per cm², find the total expenses required for polishing and painting the surface of the bookshelf.

Solution. The external dimensions of the wooden bookshelf are

Height=110 cm, breadth =25 cm and depth of the wooden bookshelf is 85 cm

Let l=85 cm,b =25 cm and h=110 cm

Since thickness of the plank is 5 cm everywhere

Internal length(L) of front portion =85 -2×5 =85 -10 =75 cm

There are 4 planks in front portion

Total internal height of three self =110 -4×5 =90 cm

Hight(H) of single self =90/3 =30 cm

Breadh(B) of each self = 25 -5 =20 cm

Total external surface area of the bookself to be polished

= TSA of whole of bookself(as a cuboid) – 3×area of open portion of the bookself

=2(lb +bh +hl) – 3(length of each self×height of each self)

=2(85×25 +25×110+110×85) – 3(75×30)

=2(2125 +2750 +9350) – 3×2250

=2×14225 – 3×6750

=28450 – 6750

=21700

External Area to be polished is 21700 cm²

Cost of polishing is 20 paise per cm²

20 paise =Rs 0.20

Therefore cost of polishing external surface area of the bookself

= 21700×0.20

=4340

So, the cost of polishing is Rs 4340

Area of the bookshelf to be painted

= 3×TSA of single shelf(as a cuboid) -3×open area of each self

=3×2(LB +BH+HL) – 3(L×H)

=6(75×20+20×30 +30×75) -3(75×30)20100-6750

=6(1500 +600 +2250) -3(2250)

=6×4350 -6750

=26100-6750

=19350

Area to be painted is 19350 cm²

The cost of painting is 10 painse per cm²

10 painse =Rs 0.10

Therefore cost of painting is 0.10 ×19350 =Rs 1935

Total expenses required for polishing and painting the surface of the bookshelf

= Cost of polishing + Cost of painting

=Rs 4340 +Rs 1935

=Rs 6275

Hence total expenses of the painting and polishing is Rs 6275

Q2.The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in fig. 13.32. Eight such spheres are used for this purpose, and are to be painted silver. Each support is a cylinder of radius 1.5cm and height 7cm and is to be painted black. Find the cost of paint required if silver paint costs 25 paise per cm² and black paint costs 5 paise per cm².

Solution. The diameter of the wooden sphere is 21 cm

The radius(r) of the wooden sphere is 21/2 =10.5 cm

Surface area of the sphere=4πr²

=4(22/7)×10.5×10.5

= 88 ×1.5×10.5

=1386

Surface area of a single sphere is 1386 cm²

Area of the sphere to be painted silver =Surface area of the sphere -area of lower circular end of the sphere attached to cynder = 1386 -πr²= 1386 -(22/7)×1.5×1.5 =1386 -7.07=1378.93

The surface area of  8 spheres to be painted =8×1378.93=11031.44 cm²

The cost of silver paint is 25 paise per cm²

35 paise =Rs 0.25

The cost of painting silver colour is

=0.25× 11031.44=Rs 2757.86

Area of the cylinder to be painted

= Curved surface area of the cylinder

=2πrh, where radius (r) of the cylinder is given 1.5 cm and height(h) is given 7 cm

=2(22/7)×1.5×7

= 44×1.5

=66

The surface area of  8 cylinders to be painted is 8×66 =528 cm²

The cost of black paint is 5 paise per cm²

5 paise =Rs 0.05

The cost of painting black colour is

=0.05×528

=Rs 26.40

The total cost of painting both colours 

=Rs 2757.86 +Rs 26.40

= Rs 2784.26

Q3.The diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease?

Solution. Let diameter of the sphere is 2r then its radius is r

Its surface area is,S =4πr²

Since diameter of the sphere is decreased by 25% then diameter of modified sphere is

=2r -25% of 2r

= 2r – r/2

= (4r -r)/2 =3r/2

So, radius of the modified sphere is 3r/4

Surface area of the modified sphere is

S’=4π(3r/4)²=9πr²/4 cm²

The curved surface area of the sphere is decreased by

S -S’ =4πr² – 9πr²/4  =7πr²/4

The percent by which the curved surface area of the sphere is decreased

=[(S-S’)/S]×100

=6.25 ×7

=43.75 %

Hence curved surface area of the sphere is decreased by 43.75 %

The volume of the hemispherical bowl is 22.46 mm³