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Question-1

A Plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is to be open at the top. Ignoring the thickness of the plastic sheet, determine:
(i) The area of the sheet required for making the box.
(ii) The cost of sheet for it, if a sheet measuring 1m2 costs
20. 

Solution:


(i) lt is given that, length(l) of box = 1.5 m
   breadth (b) of box = 1.25 m

Depth(h) of box = 65 cm = 0.65 m


The area of the sheet required for making the box = lb + 2(bh + hl)                      

                                     = (1.5)(1.25) + 2[(1.25)(0.65) + (0.65)(1.5)]

                                     = 1.875+ 2(0.8125+0.975)

                                     = 1.875 + 2(1.7875)

                                     = 1.875 + 3.575 = 5.45 m2.


(ii) Cost of sheet per m2 area =  Rs 20
     cost of sheet of 5.45 m2 area = Rs(5.45*20) = Rs 109.

Question-2

The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of 7.50 Per m2.

Solution:
Given,
l = 5 m

b = 4 m
h = 3m

Area of the walls of the room = 2(l + b) h

                                           = 2( 5 + 4) 3 = 54 m2

Area of the ceiling = lb = (5) (4) = 20 m2

Total area of the walls of the room and the ceiling = 54 m2 + 20 m2 = 74 m2


Cost of white washing the walls of the room and the ceiling = 74 × 7.50 =   555.

Question-3

The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of Rs.10 per m2 is ₹ 15000, Find the height of the hall.
[Hint: Area of the four walls = Lateral Surface area.]

Solution:
Let the length, breadth and height of the rectangular hall be l m, b m and h m respectively.
Perimeter = 250 m

2(l +b) = 250 l + b= 125           ...(1)

Area of the four walls = = 1500 m2


2(l + b) h = 1500 (l + b) h = 750
125 h = 750                        Using (1)

h = h = 6 m

Hence the height of the hall is 6 m.

Question-4

The paint in a certain container is sufficient to paint an area equal to 9.375 m2. How many bricks of dimensions 22.5 cm 10 cm 7.5 cm can be painted out of this container?

Solution:
 
For a brick
l = 22.5 cm

b= 10 cm

h = 7.5 cm

Total surface area of a brick = 2(lb + bh + hl)
= 2(22.5 × 10 + 10 × 7.5 + 7.5 × 22.5)

= 2(225 + 75 + 168.75)

= 2(468.75) = 937.5 cm2 = 0.09375 m2

Number of bricks that can be painted out = = 100.
 

Question-5

 A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10cm wide and 8 cm high.
(i) Which box has the greater lateral surface area and by how much?
(ii) Which box has the smaller total surface area and by how much?

Solution:

(i) Each edge of the cubical box (a) = 10 cm

Lateral surface area of the cubical box = 4a2 = 4(10)2 = 400 cm2.

For cuboidal box
l = 12.5 cm
b = 10 cm

h = 8 cm

Lateral surface area of the cuboidal box = 2(l + b) h = 2(12.5 + 10)(8) = 360 cm2.

Cubical box has the greater lateral surface area than the cuboidal box by
(400 - 360) cm2,

i.e., 40 cm2.

(ii) Total surface area of the cubical box = 6a = 6(10)2 = 600 cm2

Total surface area of the cuboidal box = 2(lb + bh + hl)


                                                   = 2[(12.5)(10) +)(10)(8) +(8)(12.5)]

                                                   = 2[125 + 80 + 100] = 610 cm2.

Cubical box has the smaller total surface area than the cuboidal box by (610 - 600) cm2i.e., 10 cm2.

Question-6

A small indoor greenhouse (herbarium) is made entirely of glass panes (Including base) held together with tape. It is 30 cm long,
25 cm wide and 25 cm high.
(i) What is the area of the glass?
(ii) How much of tape is needed for all the 12 edges?

Solution:
(i) For herbarium
l = 30 cm
b = 25 cm
h = 25 cm

Area of the glass = 2(lb + bh + hl)

                           = 2[(30)(25) + (25)(25) + (25)(30)]

                           = 2[750 + 625 + 750] = 4250 cm2.

(ii) The tape needed for all the 12 edges

                                      = 4(l +b + h)
                                      = 4(30 + 25 + 25) = 320 cm.

Question-7

Shanti Sweets stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25 cm 20 cm 5 cm and the smaller of dimensions 15 cm 12cm 5 cm. For all the overlaps, 5% of the total surface area is required extra. If the cost of the cardboard is 4 for 1000 cm2, Find the cost of cardboard required for supplying 250 boxes of each kind.

Solution:
For bigger box
l = 25 cm, b = 20 cm, h = 5 cm

Total surface area of the bigger box = 2(lb + bh + hl)

                      = 2[(25)(20) + (20)(5) + (5)(25)]

                      = 2[500 + 100 + 125] = 1450 cm2

Cardboard required for all the overlaps = 1450 × = 72.5 cm2
Net surface area of the bigger box = 1450 cm2 + 72.5 cm2 = 1522.5 cm2

Net surface area of 250 bigger boxes = 1522.5 × 250 = 380625 cm2

Cost of cardboard = `1522.50.
For smaller box
l =15 cm, b = 12 cm, h = 5 cm

Total surface area of the smaller box = 2(lb + bh + hl)
                                                            = 2[180 + 60 + 75]
                                                            = 630 cm2


Cardboard required for all the overlaps = 630 × = 31.5 cm2

Net surface area of the smaller box = 630 cm2 + 31.5 cm2
                                                          = 661.5 cm2


Net surface area of 250 smaller boxes = 661.5 × 250 = 165375 cm2

Cost of cardboard =  × 165375 = 661.50.

Question-8

Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5m, with base dimensions 4m 3m?

Solution:
For shelter
l = 4 m, b = 3 m, h = 2.5m
Total surface area of the shelter = lb + 2(bh + hl)
                                                = (4)(3) + 2[(3)(2.5) + (2.5(4)]
                                                            = 12 + 2[7.5 + 10] = 47 m2

Hence 47 m2 of tarpaulin will be required.

Question-9

The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder.

Solution:
Let the radius of the base of the cylinder be r cm
h = 14 cm                  (Given) 
Curved surface area = 88 cm2

2πrh = 88 = 88

r = r = 1
2r = 2
Hence the diameter of the base of the cylinder is 2 cm.

Question-10

It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?

Solution:
h = 1m = 100 cm
2r = 140 cm
r = cm = 70 cm
Total surface area of the close cylindrical tank = 2πr(h + r)

                                                                  =

                                                                   = 74800 cm2 = m2


                                                                   = 7.48 m2
Hence 7.48 square metres of the sheet are required.

Question-11

A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm. Find its
(i) inner curved surface area.
(ii) Outer curved surface area.
(iii) Total surface area.

Solution:
In the given cylinder,
h = 77 cm, 
2r = 4 cm,  r = 2 cm
2R = 4.4 cm
R = 2.2 cm
                                                                                                                        
(i) Inner curved surface area = 2π rh
= 968 cm2
(ii) Outer curved surface area = 2π Rh

                                         = = 1064.8 cm2

(iii) Total surface area = 2π Rh + 2π rh + 2π (R2 - r2)

                               = 1064.8+

                               = 1064.8 + 968 +

                               = 1064.8 + 968 + 2 x


                               = 1064.8 + 968 + 5.28 = 2038.08 cm2.

Question-12

The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m2.

Solution:
2r = 84 cm
r = 42 cm,

h = 120 cm

Area of the playground levelled in taking 1 complete revolution
= 2π rh
        = = 31680 cm2
Area of the playground = 31680 × 500

                                      = 15840000 cm2 =
                                      = 1584 m2
Hence the area of the playground is 1584 m2.

Question-13

A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate  12.50 per m2.

Solution:
In a given cylinderical pillar, Diameter = 2r = 50 cm
∴ r = 25 cm = 0.25 m
h = 3.5 m

Curved surface area of the pillar = 2πrh

                                               = = 5.5 m2

Cost of painting the curved surface of the pillar at the rate of Rs. 12.50 per m2
                                                     =  5.5 × 12.50 = 68.75

Question-14

Curved surface area of a right circular cylinder is 4.4 m2. If the radius of the base of the cylinder is 0.7 m, find its height.

Solution:
Let the height of the right circular cylinder be h m.
r = 0.7 m
Curved surface area = 4.4 m2
                       Þ 2π rh = 4.4
 Þ = 4.4
4.4 h = 4.4 h = 1 m
Hence the height of the right circular cylinder is 1 m.

Question-15

The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find
(i) its inner curved surface area.
(ii) the cost of plastering this curved surface at the rate of
 40 per m2.

Solution:
(i)Given, 2r = 3.5 m
r = 1.75 m
    h = 10 m  

Inner curved surface area of the circular well = 2πrh

                                       = = 110 m2.
(ii) Cost of plastering the curved surface at the rate of 40 per m2
                       = 110 × 40 = ₹ 4400.

Question-16

In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.

Solution:
Given,
h = 28 m
2r = 5 cm
r = = ==

Total radiating surface in the system = 2πrh
                             = = 4.4 m2.

Question-17

 Find (i) the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high.
(ii) How much steel was actually used, if of the steel actually used was wasted in making the tank.

Solution:
(i) 2r = 4.2 m
r = = 2.1 m

h = 4.5 m
Lateral or curved surface area = 2πrh

                = = 59.4 m2

(ii) Total surface area = 2π r (h + r)

             =

             =

             = 87.12 m2

Let the actual area of steel used be x m2.
Since of the actual steel used was wasted, the area of the steel which has gone into the tank = of x

= 87.12

x = = 95.04 m2

Steel actually used = 95.04 m2.

Question-18

In figure, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20cm and height of 30cm. A margin of 2.5cm is to be given for folding it over the top and bottom of the frame. Find how much cloth is required for covering the lampshade.

Solution:
Given, 2r = 20 cm
r = 10 cm

h = 30 cm


Cloth required = 2πr(h + 2.5 + 2.5)

                          = 2πr(h + 5) =
                          = 2200 cm2

Question-19

The students of a vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius 3 cm and height 10.5 cm. The vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was required to be bought for the competition?

Solution:
Given, r = 3 cm
h = 10.5 cm
Cardboard required for 1 competitor = 2πrh + πr2

 
                = 198+ =

                = 
Cardboard required for 35 competitors to be bought for the competition = 35 × = 7920 cm2
Hence 7920 cm2 of cardboard was required to be bought for the competition.

Question-20

Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.

Solution:
Diameter of the base = 10.5 cm
Radius of the base (r) = cm = 5.25 cm
Slant height (l) = 10 cm

Curved surface area of the cone = πrl

                         = = 165 cm2.

Question-21

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24m.

Solution:
Slant height (l) = 21 m

Diameter of base = 24 m

Radius of base (r) = m = 12 m


Total surface area of the cone = πr(l + r)

                      =

                      = =

                       = 1244 m2.

Question-22

Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find

(i) radius of the base and

(ii) Total surface area of the cone.

Solution:
(i) Slant height (l) = 14 cm

Curved surface area = 308 cm2
                                Þ π rl = 308


            Þ = 308
                                     Þ r =

                    Þ r = 7 cm
Hence the radius of the base is 7 cm. 
(ii) Total surface area of the cone = π r(l + r)
                                                   =

                                                   = = 462 cm2

Hence the total surface area of the cone is 462 cm2.

Question-23

A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) Slant height of the tent.
(ii) Cost of the canvas required to make the tent, if the cost of 1 m2 canvas is
 70.

Solution:
(i) Given, h = 10 m, r = 24 m

∴ l = =

     =  =

     = 26 m

Hence the slant height of the tent is 26 m.

(ii) Curved surface area of the tent = πrl = m2

Cost of the canvas required to make the tent, if the cost of 1 m2 canvas is  70

                                             =  137280.

Hence the cost of the canvas is
137280.

Question-24

What length of tarpaulin 3m wide will be required to make conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20cm (use = 3.14).

Solution:
For conical tent
Given,
h = 8 m, r = 6 m
l =

     = = = = 10 m

Width surface area = πrl = 3.14 × 6 × 10 = 188.4 m2

Width of tarpaulin = 3 m

Length of tarpaulin = = 62.8 m
Extra length of the material required = 20 cm = 0.2 m

Actual length of tarpaulin required = 62.8 m + 0.2 m = 63 m.

Question-25

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white - washing its curved surface at the rate of Rs. 210 per 100 m2.

Solution:
Slant height (l) = 25 m

Base diameter = 14 m

Base radius (r) = = 7 m


Curved surface area of the tomb = π rl = =550 m2

Cost of white-washing the curved surface of the tomb at the rate of Rs. 210 per 100 m2

                                                      = Rs. = Rs. 1155.

Question-26

 A Joker’s cap is in the form of a right circular cone of base radius 7cm and height 24cm. Find the area of the sheet required to make 10 such caps.

Solution:
Base radius (r) = 7 cm

Height (h) = 24 cm


Slant height (l) = = = = = 25 cm

Curved surface area of a cap = π rl = = 550 cm2

Curved surface area of 10 caps = 550 × 10 = 5500 cm2
Hence the area of the sheet required to make 10 such caps is 5500 cm2.

Question-27

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1m. If the outer side of each of the cones is to be painted and the cost of painting is Rs.12 per m2, What will be the cost of painting all these cones? (use = 3.14 and take = 1.02).

Solution:
Given, Base diameter of the cone = 40 cm
Base radius (r) = cm = 20 cm = m = 0.2m
Height (h) = l m
l = = = = = 1.02 m (approximately).

Curved surface area = π rl = 3.14 × 0.2 × 1.02 = 0.64056 m2
Curved surface area of 50 cones = 0.64056 × 50 m2 = 32.028 m2

Cost of painting all these cones = 32.028 × 12 = 384.336 = Rs. 384.34 (approximately).

Question-28

Find the surface area of a sphere of radius: (i) 10.5 cm (ii) 5.6 cm (iii) 14cm

Solution:
(i) r = 10.5 cm Surface area = 4π r2 = 4 x = 1386 cm2.

(ii) r = 5.6 cm

Surface area = 4π r2 = 4 x = 394.24 cm2.

(iii) r = 14 cm

Surface area = 4π r2 = 4 x = 2464 cm2.

Question-29

Find the surface area of a sphere of diameter: (i) 14 cm (ii) 21cm (iii) 3.5m

Solution:
(i) Diameter - 14 cm

Radius (r) = cm = 7 cm

Surface area = 4π r2 = 4 x = 616 cm2.

(ii) Diameter = 21 cm

Radius (r) = cm

Surface area = 4π r2 = = 1386 cm2.

(iii) Diameter = 3.5 cm


Radius (r) = cm = 1.75 cm

Surface area = π r2 = = 38.5 cm2.

Question-30

Find the total surface area of a hemisphere of radius 10cm. (use = 3.14).

Solution:
r = 10 cm.

Total surface area of the hemisphere = 3π r2 = 3 × 3.14 × (10)2=942 cm2.

Question-31

The radius of a spherical balloon increases from 7cm to 14cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.

Solution:

Case I. r = 7 cm

Surface area = 4π r2 = = 616 cm2.

Case II. r = 14 cm

Surface area = 4π r2 = = 2464 cm2

Ratio of surface areas of the balloon = 616 : 2464

= = = 1 : 4

Question-32

A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs.16 per 100cm2.

Solution:

Inner diameter = 10.5 cm
Inner radius (r) = cm = 5.25 cm
Inner surface area = 2πr2 = = 173.25 cm2
Cost of tin-plating at the rate of Rs. 16 per 100 cm2
                             = 173.25× Rs. 27.72

Question-33

Find the radius of a sphere whose surface area is 154cm2.

Solution:

Let the radius of the sphere be r cm.
Surface area = 154 cm2
4πr2 = 154
= 154

r2 =

r2 =

r = r = = 3.5 cm
Hence the radius of the sphere is 3.5 cm.

Question-34

The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.

Solution:
 
Let the diameter of the earth be 2r.

Then diameter of the moon = (2r) =


Radius of the earth = = r

and, Radius of the moon = =

Surface area of the earth = 4π r2
and, Surface area of the moon = 4π  (2 =π r2 

Ratio of their surface areas =    
       

                                          = =

∴ Ratio of their surface areas  = 1 : 16

Question-35

A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.

Solution:

Inner radius of the bowl = 5 cm
Thickness of steel = 0.25 cm
Outer radius of the bowl = 5 + 0.25 = 5.25 cm
Outer curved surface of the bowl = 4π r2
               = = 346.5 cm2.

Question-36

A right circular cylinder just encloses a sphere of radius r. Find

(i) surface area of the sphere

(ii) curved surface area of the cylinder

(iii) ratio of the areas obtained in (i) and (ii).

Solution:
 
(i) Surface area of the sphere = 4πr2

(ii) For cylinder

 


Radius of the base = r
Height = 2r
Curved surface area of the cylinder = 2π (r)(2r) = 4π r2


(iii) Ratio of the areas obtained in (i) and (ii)
=

                                                                   = =

                                                                    = 1 : 1.

Question-37

A match box measures 4 cm ×2.5 cm × 1.5 cm. What will be the volume of a packer containing 12 such boxes?

Solution:

Volume of a matchbox = 4 × 2.5 × 1.5 cm2 = 15 cm2

Volume of a packet containing 12 such boxes = 15 × 12 cm2 = 180 cm2

Question-38

A cuboidal water tank is 6m long, 5m wide and 4.5 m deep. How many litres of water can it hold?
(Hint.1 m3 = 1000 l)

Solution:

Capacity of the tank = 6 × 5 × 4.5 m3 = 135 m3

Volume of water it can hold = 135 m3 = 135 × 1000 l =135000 l.

Question-39

A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?

Solution:

Let the height of the cuboidal vessel be h m.
l = 10m

b = 8m

Capacity of the cuboidal vessel = 380 m3
lbh = 380 (10)(8)h = 380

h = h =

h = 4.75 m
Hence the cuboidal vessel must be made 4.75 m high.

Question-40

Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of Rs. 30 per m3.

Solution:
l = 8 m, b = 6 m, h = 3 m
Volume of the cuboidal pit = lbh = 8 × 6 × 3 m3 = 144 m3

Cost of digging the cuboidal pit @ Rs. 30 per m3 = Rs. 144 × 30 = Rs. 4320.

Question-41

The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10m.

Solution:

Let the breadth of the cuboidal tank be b m.
l = 2.5m
h = 10 m

Capacity of the cuboidal tank = 50000 litres = m3 = 50 m3

lbh = 50 2.5 × b × 10 = 50

25b = 50 b = = 20 m
Hence the breadth of the cuboidal tank is 20 m.

Question-42

A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20m × 15m × 6m. For how many days will the water of this tank last?

Solution:

Requirement of water per head per day = 150 litres

Requirement of water for the total population of the village per day

                      = 150 × 4000 litres
                      = 600000 litres
                      = m3 = 600 m3
For tank
l = 20m
b=15m
h = 6m

Capacity of the tank = 20 × 15 × 6 m3 = 1800 m3

Number of days for which the water of this tank last


                           =

                           = = 3

Hence the water of this tank will last for 3 days.

Question-43

A godown measures 40m × 25m × 10m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25m × 0.5m that can be stored in the godown.

Solution:
For godown

l = 40m, b = 25 m, h= 10 m
Capacity of the godown = lbh = 40 × 25 × 10m3 =10000m3

For a wooden crate

l =1.5m, b= 1.25m, h = 0.5 m

Capacity of a wooden crate = lbh = 1.5 × 1.25 × 0.5 m3 = 0.9375 m3
We have,


∴ Maximum number of wooden crates that can be stored in the godown == 10666.66

Hence the maximum number of wooden crates that can be stored in the godown is 10666.

Question-44

A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, Find the ratio between their surface areas.

Solution:

Side of the solid cube (a) = 12 cm

Volume of the solid cube = a3

                                      = (12)3 =12 × 12 × 12 cm3 = 1728 cm3
It is cut into eight cubes of equal volume.
Volume of a new cube = cm3 = 216 cm3
Let the side of the new cube be x cm.
Then, volume of the new cube = x3 cm3.
According to the question,
    x3 = 216

x = (216)1/3 x = (6 × 6 × 6)1/3

x = 6 cm

Hence the side of the new cube will be 6 cm.

Surface area of the original cube = 6x2 = 6(12)2 cm2

Surface area of the new cube = 6x2 = 6(6)2 cm2


Ratio between their surface areas =

= = = 4 : 1

Hence the ratio between their surface areas is 4 : 1.

Question-45

A river 3 m deep and 40 m wide is flowing at the rate of 2 Km per hour. How much water will fall into the sea in a minute?

Solution:
In one hour
l = 2 km = 2 × 1000 m = 2000 m

b = 40 m

h = 3 m

Water fell into the sea in one hour = lbh = 2000 × 40 × 3 m3

Water fell into the sea in a minute = 4000 m3

Hence 4000 m3 of water will fall into the sea in a minute.

Question-46

The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? 
(1000 cm3 =1 l )

Solution:
Let the base radius of the cylindrical vessel be r cm
Then, circumference of the base of the cylindrical vessel = 2πr cm
According to the question,
2π r = 132

= 132 r =

r = 21 cm; h = 25 cm
Capacity of the cylindrical vessel = πr2h

                                                     = cm3

                                                     = 34650 cm3 = = 34.65 l
⇒ Therefore,such vessel can hold 34.65 liters of water.

Question-47

The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, If 1 cm3 of wood has a mass of 0.6 g.

Solution:

Inner diameter = 24 cm

Inner radius (r) = cm = 12 cm

 Outer diameter = 28 cm
Outer radius (R) = cm = 14 cm
Length of the pipe (h) = 35 cm

Outer volume = π R2h

            = = 21560 cm3
Inner volume = π r2h
           = = 15840 cm3
Volume of the wood used = Outer volume - Inner volume
                                           = 21560 cm3 - 15840 cm3                                                  
                                           = 5720 cm3    
Mass of the pipe = 5720 × 0.6 g = 3432 g = 3.432 kg.

Question-48

A soft drink is available in two packs –
(i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and
(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?

Solution:
 
(i) For tin can
l = 5 cm

b = 4 cm

h =15 cm

Capacity = l × b × h = 5 × 4 × 15 cm3 = 300 cm3.

(ii) For plastic cylinder

Diameter = 7 cm

Radius (r) = cm
Height (h) = 10 cm
Capacity = π r2h
                 = = 385 cm3
Clearly the second container i.e., a plastic cylinder has greater capacity than the

first container i.e., a tin can by 385 – 380 = 5 cm3.

Question-49

If the lateral surface of a cylinder is 94.2 cm2 and its height is 5 cm, then find
(i) radius of its base (ii) its volume (use π = 3.14)

Solution:
(i) Let the radius of the base of the cylinder be r cm.
h = 5 cm
Later surface = 94.2 cm2
          Þ 2π rh = 94.2 2 × 3.14 × r × 5 = 94.2
r = r =
r = 3 cm
Hence the radius of the base is 3 cm. 
(ii) r = 3 cm, h = 5 cm
Volume of the cylinder = π r2h = 3.14 × (3)2 × 5 = 141.3 cm3.

Question-50

It costs Rs.2200 to paint the inner curved surface of a cylindrical vessel 10m deep. If the cost of painting is at
the rate of Rs. 20 per m2, Find
(i) inner curved surface area of the vessel,
(ii) radius of the base.
(iii) Capacity of the vessel.

Solution:
 
(i) Inner curved surface area of the vessel = = 110 m2

(ii) Let the radius of the base be r m, 
     h = 10 m
Inner curved surface area = 110 m2
2π rh = 110 = 110

r = r =

r = 1.75 m
Hence the radius of the base is 1.75 m.
(iii) r = 1.75 m
      h = 10 m
Capacity of the vessel = π r2h
                                      =
                                      = 96.25 m3
Hence the capacity of the vessel is 96.25 m3(or 96.25 kl).

Question-51

The capacity of a closed cylindrical vessel of height 1m is 15.4 litres. How many square metres of metal sheet would be needed to make it?

Solution:
In a cylinderical vessel,
h = 1 m
Capacity = 15.4 litres = = 0.0154 m3


Let the radius of the base be r m.

Capacity = 0.0154 m3

  Þ π r2h = 0.0154

= 0.0154

r2 =

r2 = 0.0049

r = r = 0.07 m
Curved surface area = 2π rh + 2π r2 =
                                                   = 0.44 + 0.0308 = 0.4708 m2
Hence 0.47 m2 of metal sheet should be needed.

Question-52

A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.

Solution:
For solid cylinder of graphite Radius (r) =
Length of the pencil (h) = 14 cm = 140 mm

Volume of the graphite = π r2h = = 110 mm3

                                                   = cm3 = 0.11 cm3
For cylinder of wood
Diameter = 7 mm
Radius(R) =
Length of the pencil (h) = 14 cm = 140 mm

Volume of the wood = π (R2 - r2)h =140

                                                         = 5280 mm3

                                                         = cm3

                                                         = 5.28 cm3

Question-53

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of
4 cm, how much soup the hospital has to prepare daily to serve 250 patients?

Solution:
cylindrical bowl 
Diameter = 7 cm Radius (r) =
Height (h) = 4 cm
Volume of soup in the cylindrical bowl = π r2h = = 154 cm3

Volume of soup to be prepared daily to serve 250 patients
                                  = 154 × 250 cm3 = 38500 cm3 (or 38.5l)
Hence the hospital has to prepare 38500 cm3(or 38.5l) of soup daily to serve 250 patients.

Question-54

Find the volume of the right circular cone with

(i) radius 6 cm, height 7cm (ii) radius 3.5 cm, height 12 cm

Solution:

i) r = 6 cm, h =7 cm
Volume of the right circular cone = =

(ii) r = 3.5 cm
h = 12 cm
Volume of the right circular cone = = = 154 cm3
 

Question-55

Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm (ii) height 12 cm, slant height 13 cm

Solution:
(i) r = 7 cm, l = 25 cm
    r2 + h2 = l2
(7)2 + h2 = (25)2 h2 = (25)2 – (7)2
h2 = 625 – 49 h2 = 576
h = h = 24 cm

Capacity = = = 1232 cm3 = 1.232 l.

(ii) h = 12 cm,
l = 13 cm
     r2+h2 = l2
r2+(12)2 = (13)2 r2 + 144 = 169
r2 = 169 – 144 r2 = 25
r = r = 5 cm

Capacity =

                  = =  
 

Question-56

The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (use p = 3.14)

Solution:
Let the radius of the base of the cone be r cm.
h = 15 cm
volume = 1570 cm3

= 1570 = 1570

r2 = r2 = 100

r = r = 10 cm

Question-57

If the volume of a right circular cone of height 9cm is 48 cm3, find the diameter of its base.

Solution:

Let the radius of the base of the right circular cone be r cm.
h = 9 cm
volume = 48π cm3
= 48π = 48

= 48 r2 =

r2 = 16 r = = 4 cm

2r = 2(4) = 8 cm
Hence the diameter of the base of the right circular cone is 8 cm.

Question-58

A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Solution:
Given, 
Diameter = 3.5 cm
Radius (r) = = 1.75 m
Depth (h) = 12 m
Capacity of the conical pit =

                                           =

                                           = 38.5 m3 = 38.5 × 1000 l

                                           = 38.5 kl.

Question-59

The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find

(i) height of the cone

(ii) slant height of the cone

(iii) curved surface area of the cone.

Solution:
In a right circular cone,

(i) Diameter of the base = 28 cm


Radius of the base (r) = = 14 cm

Let the height of the cone be h cm.
Volume = 9856 cm3

= 9856 = 9856

h = h = 48 cm

Hence the height of the cone is 48 cm.

(ii) r = 14 cm, h = 48 cm

l = = = = = 50 cm

Hence the slant height of the cone is 50 cm.

(iii) r = 14 cm

l = 50 cm
Curved surface area = π rl = = 2200 cm2

Hence the curved surface area of the cone is 2200 cm2.

Question-60

A right triangle ABC with sides 5 cm, 12 cm and 13 cm is resolved about the side 12 cm. Find the volume of the solid so obtained.

Solution:
The solid obtained will be a right circular cone whose radius of the base is 5 cm and height is 12 cm.
r = 5 cm, h = 12 cm
                                                                                                
Volume = = = 100π cm3

Hence the volume of the solid so obtained is 100π cm3

Question-61

If the triangle ABC with sides 5 cm, 12 cm, 13 cm is revolved about the side 5cm, then find the volume of the solid so obtained. Find also the volume of the solid is revolved about the side 12cm and the ratio of the volumes of the two solids obtained

Solution:

The solid obtained will be a right circular cone whose radius of the base is 12 cm and height is 5 cm.
r = 12 cm, h = 5 cm

Volume = = = 240π cm3 
 
The solid obtained will be a right circular cone whose radius of the base is 5cm and height is 12 cm.
 r = 5 cm, h = 12 cm

Volume = = = 100π cm3   
Ratio of the volumes of the two solids obtained = 240π cm3 : 100π cm3  
                                                                                = 12 : 5

Question-62

A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.

Solution:
 
For heap of wheat
Diameter = 10.5 m

Radius (r) = cm = 5.25 m

Height (h) = 3m
Volume = = = 86.625 m3

Slant height (l) = = = 6.05 m (approx.)

Curved surface area = πrl = = 99.825 m2

Hence the area of the canvas required is 99.825 m2

Question-63

Find the volume of a sphere whose radius is (i) 7 cm (ii) 0.63 m.

Solution:

(i) r = 7 cm
Volume of the sphere = = = = 1437
(ii) r = 0.63 m
volume of the sphere = = = 1.05 m3(approx.)

Question-64

Find the amount of water displaced by a solid spherical ball of diameter (i) 28 cm (ii) 0.21 m

Solution:
 
(i) Diameter = 28 cm

Radius (r) = cm = 14 cm

Amount of water displaced = = = cm3 = 11498

(ii) Diameter = 0.21 m

Radius (r) = = 0.105 m

Amount of water displaced = = = 0.004581 m3

Question-65

The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g percm3?

Solution:

Diameter (r) = = 2.1 cm

Volume = = = 38.808 cm3

Density = 8.9 g per cm3

Mass of the ball = Volume × Density = 38.808 × 8.9 = 345.39 (approx.)

Question-66

The diameter of the moon is approximately one–fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Solution:

Let the radius of the earth = r
Then, diameter of the earth = 2r

Diameter of the moon = =

Radius of the moon = =

Volume of the earth (v1) =

Volume of the moon(v2) = =  

                                 = (volume of the earth)

Question-67

How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold?

Solution:

Diameter = 10.5 cm

Radius (r) = = 5.25 cm

Amount of milk = =

                            = 303 cm3 (approx.) = 0.303 litres (approx).

Question-68

A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1m, then find the volume of the iron used to make the tank.

Solution:
For hemispherical tank,
Inner radius (r) = 1m

Thickness of iron sheet = 1 cm = 0.01 m
Outer radius (R) = Inner radius (r) + Thickness of iron sheet
                            = 1.01 m
Volume of the iron used to make the tank =

                                                                   =   

                                                                  = 0.06348 m3 (approx.)

Question-69

Find the volume of a sphere whose surface area is 154 cm2.

Solution:

Let the radius of the sphere be r cm.

Surface area = 154 cm2 4π r2 = 154 = 154
 

= =
 

h = h = cm

Volume of the sphere = = = = 179 cm3.

Question-70

A dome of a building is in the form of a hemisphere. From inside, it was white-Washed at the cost of Rs. 498.96. If the cost of white-washing is Rs. 2.00 per square metre, Find the

(i) Inside surface area of the dome

(ii) Volume of the air inside the dome.

Solution:

(i) Inside surface area of the dome = = 249.48 m2

(ii) Let the radius of the hemisphere be r m.

Inside surface area = 249.48 m3


2π r2 = 249.48

= 249.48

=

= 39.69

 r =

 r =6.3 cm
Volume of the air inside the done = = = 523.9 m3 (approx.)

Question-71

Twenty seven solid iron spheres, each of radius r and surface area s are melted to form a sphere with surface area S’. Find the
(i) radius r’ of the new sphere

(ii) ratio of S and S.

Solution:

(i) Volume of a solid iron sphere =

Volume of 27 solid iron spheres = 27

Volume of the new sphere = 36π r3
Let the radius of the new sphere be r’.
Then,
Volume of the new sphere =
According to the question,
= 36π r’3

r’3 = r’3 = 27r3

r’ = (27r3)1/3 r’ = (3 × 3 × 3 × r3)1/3
r’ = 3r
Hence the radius r’ of the new-sphere is 3r.
(ii) S = 4π r2

S’ = 4π (3r)2

= = = 1 : 9

Hence the ratio of S and S’ is 1 : 9

Question-72

A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to fill this capsule?

Solution:

Diameter of the capsule = 3.5 mm
 

Radius of the capsule (r) = = 1.75 mm
 

Capacity of the capsule = = = 22.46 mm3 (approx.)

Hence 22.46 mm3 (approx.) of medicine is needed to fill this capsule.

Question-73

The outer diameter of a spherical shell is 10 cm and the inner diameter is 9 cm. Find the volume of the metal contained in the shell

Solution:
Outer diameter = 10 cm

Outer radius (R) = 10/2 cm = 5 cm

Inner diameter = 9 cm
Inner radius (r) = cm
Volume of the metal contained in the shell =
 
                                       = =

                                       =

                                       = =

Question-74

If the number of square centimetres on the surface of a sphere is equal to the number of cubic centimetres in its volume, what is the diameter of the sphere?

Solution:

Let the radius of the sphere be r cm. Then,
Surface area = 4π r2 cm2
and, volume
According to the question, 4π r2 =
r = 3
2r = 6
Hence the diameter of the sphere is 6 cm.

Question-75

A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.

Solution:

Volume of the cone = cm3

Volume of the hemisphere =

According to the question, =
h = 2r
Height of the cone = 2r cm.
Height of the hemisphere = r cm.
Ratio of their heights = 2r : r = 2 : 1




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