NCERT Exemplar Class 9 Maths Solutions Chapter 13 Surface Areas and Volumes

Question:1
The radius of a sphere is 2r, then its volume will be
(A) 4/3 πr³
(B) 4 πr³
(C) 8/3 πr³
(D) 32/3 πr³

Answer (D)

We know that the volume of a sphere is given as 4/3 πr³, where r is the radius of the sphere.

In this question, it is given that the Radius of the sphere is 2r.
So, the Volume of the sphere will be equal to:
= 4/3 π(2r)³
= (4×8)/3 π(r)³
= 32/3 πr³
So, option (D) is the correct.

Question:2

The total surface area of a cube is 96cm3. The volume of the cube is:
(A) 8cm3
(B) 512cm3
(C) 64cm3
(D) 27cm3

Answer (C)

We know that the total surface area of cube is given as 6a2 (where a is the side length of the cube)
Here we have been given that total surface area =96cm2
So, 6a2 = 96cm2
a2=966=16
a=4cm
Now we know that the volume of a cube =(sidelength)3
Volume =a3=(4)3=64cm3
So, option (C) is the correct answer.

Question:3

A cone is 8.4cm high and the radius of its base is 2.1cm. It is melted and recast into a sphere. The radius of the sphere is :
(A) 4.2cm
(B) 2.1cm
(C) 2.4cm
(D) 1.6cm

Answer (B)
In the question it i given that the height of cone =8.4cm
And, Radius of cone =2.1cm
The cone is melted and recast into a sphere
So,
Volume of sphere = Volume of cone
43πR3=13πr2h
(where R is the radius of sphere, r is the radius of cone, h is the height of cone)
R3=r2h4
Putting the given values,
R3=(2.1)2×(8.4)4=(2.1)2×2.1
R3=(2.1)3
R=2.1cm
Hence, option (B) is the correct answer.

Question:4

In a cylinder, radius is doubled and the height is halved, the curved surface area will be
(A) halved
(B) doubled
(C) same
(D) four times

Answer (C)
We know that the curved surface area of a cylinder is 2πrh
(Where r is radius and h is height)
Let, original radius =r
Original heigth = h
Original curved surface area (C1)=2πrh
Now it is given that radius is doubled and height is halved
New radius (R)=2r
New height (H) =h2
So, new curved surface area (C2)=2πRH
Now, C2=2πRH
C2=2π(2r)×h2
=2πrh
C2=2πrh
then
C2=C1
So, option (C) is the correct answer.

Question:5

The total surface area of a cone whose radius is r2 and slant height 2l is
(A) 2πr(l+r)
(B) πr(l+r4)
(C) πr(l+r)
(D) 2πrl

Answer (B)
We know that total surface area of cone is given as =π.RL+πR2
Where R is the radius of its base and L is the slant height.
Now, it is given that :
Radius of cone (R)=r2
Slant height (L) =2l
So total surface area =π(r2)(2l)+π(r2)2
=πrl+πr24
=πr(l+r4)
So, option (B) is the correct answer.

Question:6

The radii of two cylinders are in the ratio of 2:3 and their heights are in the ratio of 5:3. The ratio of their volumes is:
(A) 10:17
(B) 20:27
(C) 17:27
(D) 20:37

Answer (B)
We know that volume of a cylinder is given as πr2h
Where r is the radius of its base and h is the height.
Now it is given that ratio of Radius =r1:r2=2:3
And, Ratio of Heights =h1:h2=5:3
So, Ratio of volumes,
πr12h1:πr22h2
=πr12h1πr22h2
=(r1r2)2(h1h2)
=(23)2(53)
=(49)(53)
=2027
=20:27
So, option (B) is the correct answer.

Question:7

The lateral surface area of a cube is 256 m². The volume of the cube is
(A) 512m³
(B) 64m³
(C) 216m³
(D) 256m³

Answer (A)
We know that Lateral surface area of cube is given as: 4(a)2
Where 'a' is the edge (side) length of the cube.
Given that lateral surface area of cube =256m2
Now,
4(a)2=256
(a)2=64
a=8m
Then
Volume of cube = (edge length)3
Volume of cube = (a)3
Volume of cube = (8)3
Volume of cube = 512 m3
So, option (A) is the correct answer.

Question:8

The number of planks of dimensions (4m×50cm×20cm) that can be stored in a pit which is 16m long, 12m wide and 4m deep is
(A) 1900
(B) 1920
(C) 1800
(D) 1840

Answer (B)
We can see that both the plank and pit will be in the form of a cuboid.
Volume of a cuboid =l×b×h
Where l is its length, b is breadth and h is height.
Given dimensions of plank =(4m×50cm×20cm)
We know that, 1m=100cm
So, Dimension of plank =(4m×0.5m×0.2m)
Volume of plank=4m×0.5m×0.2m=4m3
Now, Dimensions of pit =(16m×12m×40m)
Volume of pit=16m×12m×40m=7680m3
Thus number of planks that can be fitted into the pit =Volume of PitVolume of plank
=76804
=1920
This is option (B) is the correct.

Question:9

The length of the longest pole that can be put in a room of dimensions (10m×10m×5m) is
(A) 15m
(B) 16m
(C) 10m
(D) 12m

Answer (A)
Dimensions of room =10m×10m×5m
We can see that the room is in form of a cuboid.
Longest side of a cuboid is its diagonal.
So, we can say that the longest pole will be faced along its diagonal.
Diagonal of cuboid =l2+b2+h2
=102+102+52
=225
=15m
So, option (A) is the correct answer.

Question:10

The radius of a hemispherical balloon increases from 6cmto12cm as air is being pumped into it. The ratios of the surface areas of the balloon in the two cases is:
(A) 1:4
(B) 1:3
(C) 2:3
(D) 2:1

Answer (A)
We know that for a hemisphere
Total surface area of =2πr2+πr2=3πr2 ( Where r is the radius)

Now it is given that the radius of a hemispherical balloon increases from 6 cm to 12 cm
So,
When r=6, total surface area (S1)=3π(6)2=36
When r=12, total surface area (S2)=3π(12)2=144
Hence, required ratio=(S1):(S2)
=36:144
=1:4
So, option (A) is the correct answer.

Question:1

Directions: Write True or False and justify your answer in each of the following :
The volume of a sphere is equal to two-thirds of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.

Answer: True
We know that the volume of the sphere =43πr3 (where r is the radius)
And Volume of cylinder =πR2h (where R is the radius and h is the height)
It is given that the height and diameter of the cylinder are equal to the diameter of the sphere
(So, radius of cylinder = r and h = 2r)
So, Volume of cylinder =πr2.2r
Volume of sphere=43πr3=23πr2(2r)=23 volume of cylinder
Therefore, the given statement is True.

Question:2

Directions: Write True or False and justify your answer in each of the following :
If the radius of a right circular cone is halved and the height is doubled, the volume will remain unchanged.

Answer: False
We know that Volume of a cone =13πr2h (where r is the radius and h is height)
Now, as the radius of a right circular cone is halved and the height is doubled,
Then new volume=13π(r2)2.2h=12(13πr2h)
=12 (original volume of cone )
Thus, the statement is false.

Question:3

Directions: Write True or False and justify your answer in each of the following :
In a right circular cone, height, radius and slant height do not always be sides of a right triangle.

Answer: False
Consider a right circular cone, with height = h, radius = r, slant height = l

We know that, in a right triangle, one angle is equal to 90°, Using Pythagoras' theorem, we know that:
(hypotenuse2=base2+height2)
l2=h2+r2
Thus, the statement is False.

Question:4

Directions: Write True or False and justify your answer in each of the following :
If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved.

Answer: True
We know that the curved surface area of the cylinder is given as 2πrh
(where r is the radius and h is the height)
Given that the radius of a cylinder is doubled, and its curved surface area is not changed
For the new cylinder,
New radius r1=2r
Let the new height be h1 cm
Now, according to the question –
2πrh=2πr1h1
2πrh=2π.2r.h1
⇒h1=h2
Hence, the height is halved.
Thus, the statement is true.

Question:5

Directions: Write True or False and justify your answer in each of the following :
The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.

Answer: True
We know that
Volume of cone =13πr2h (where r is the radius and h is height)
Volume of the hemisphere =23πr3 (where r is the radius)
For largest right circular cone that can be fitted in a cube whose edge is 2r, the height of cone will be 2r and its diameter will also be equal to 2r. So, radius of its base is equal to r.
So, volume of cone =13πr2(2r)=23πr3
= volume of hemisphere
Thus, the statement is true.

Question:6

Directions: Write True or False and justify your answer in each of the following :
A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone.

Answer: True
We know that
Volume of cylinder =πR2H (where R is the radius and H is the height)
Volume of cone =13πr2h (where r is the radius and h is the height)
Given that a cylinder and a right circular cone have the same base and height.
So let r be the radius and h be the height of both.
Now, the Volume of the cylinder will be, V1=πr2h
And the Volume of the cone will be, V2=13πr2h
So we get:
∴V2=13V1
Therefore, V1=3V2
Hence, the statement is true.

Question:7

Directions: Write True or False and justify your answer in each of the following :
A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1:2:3.

Answer: True
We have been given that a cone, a hemisphere and a cylinder stand on equal bases and have the same height.
Let the radius of the base be r and the height be h.
Now we know that
Volume of cone, V_Cone= 13πr2h....(i)
Volume of hemisphere, V_Hemisphere= 23πr2h....(ii)
Volume of cylinder, V_Cylinder= πr2h....(iii)
From equations (i) (ii) and (iii)
V_Cone:V_Hemisphere:V_Cylinder
=13πr2h:23πr2h:πr2h
=13:23:1
=1:2:3
Thus, the statement is true.

Question:8

Directions: Write True or False and justify your answer in each of the following :
If the length of the diagonal of a cube is 63 cm, then the length of the edge of the cube is 3 cm.

Answer: False
We know that the diagonal of the cube with side a is given as a3
Now in the question, the diagonal of the cube is 63 cm
So we can write
⇒ a3 = 63
Which gives,
a=6cm
So the side of the cube is 6 cm.
Thus, the statement is false.

Question:9

Directions: Write True or False and justify your answer in each of the following :
If a sphere is inscribed in a cube, then the ratio of the volume of the cube to the volume of the sphere will be 6:π.

Answer: True
Explanation:
Volume of cube (V1)=( Side )3 Radius of sphere (r)= Side 2 Volume of sphere (V2)=43π(r3)=43π( Side 2)3=43π×( Side )38V1V2=( side )343π( side )38=6πV1:V2=6:π

Question:10

Directions: Write True or False and justify your answer in each of the following:
If the radius of a cylinder is doubled and the height is halved, the volume will be doubled.

Answer: True
The given statement is: If the radius of a cylinder is doubled and the height is halved, the volume will be doubled.
We know that the volume of a cylinder is given as πr2h
where the radius of the cylinder is r and the height is h.
Let the original volume be,
V1=πr2h
Now the radius of a cylinder is doubled and the height is halved,
So,
New Radius (R) = 2r
New Height (H)=h2
Then, a new Volume of the cylinder
V2=π(2r)2h2
V2=2πr2h=2×V1
Hence, the given statement is true.

Question:1

Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimensions 16cm×8cm×8cm. When 16 spheres are packed the box is filled with preservative liquid. Find the volume of this liquid. Give your answer to the nearest integer. [Useπ=3.14]

Answer: 488
We know that the volume of a sphere is given as: 4/3 πr³ (where r is the radius)
Given that the radius of each sphere (r) = 2cm
So, Volume of sphere = 4/3 πr³
=43×π×(2)3
=323×3.14
Volume of 16 spheres =16×323×3.14
=535.89cm3
Internal dimensions of given rectangular box =16×8×8cm
The box is in the form of a cuboid.
Volume of rectangular box =l×b×h
=16cm×8cm×8cm
=1024cm3
Now it is given that when 16 spheres are packed, the box is filled with preservative liquid. So,
The total volume of the rectangular box = Volume of 16 spheres - Volume of preservative liquid
The volume of preservative liquid =(1024−535.89) cm³
=488.11cm3
Thus, the volume of preservative liquid is 488.11 cm³.

Question:2

A storage tank is in the form of a cube. When it is full of water, the volume of water is 15.625 m3. If the present depth of water is 1.3m, find the volume of water already used from the tank.

Answer: 8.125 m³
In the question, it is given that:
Volume of water in tank=15.625m3
Depth of water =1.3m
Now we know that,
Volume of cube is given as a3 (where a is the side length)
a3=15.625
a=15.6253
a=2.5m
The present depth of water volume is given as 1.3m.
Height of water used = 1.2 m
Volume of water already used from the tank
= area of base of the cube × Height of water used
=a2×1.2
=(2.5)2×1.2
=6.25×1.2
=7.5m3
Hence volume of water already used from the tank is 7.5m3.

Question:3

Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is completely immersed in water.

Answer: 38.808 cm3
Given that a solid spherical ball of diameter (d) 4.2 cm is completely immersed in water.
Radius of ball=d2
=4.22=2.1cm
Now we know that,
Volume of the spherical ball =43πr3 (where r is the radius)
=43×π×(2.1)3
=43×π×(9.261)
=43×227×9.261
=38.808cm3
Hence, volume of the spherical ball = 38.808 cm3
The amount of water displaced is the same as the volume of the spherical ball.
So we have the amount of water displaced = 38.808 cm3.

Question:4

How many square metres of canvas is required for a conical tent whose height is 3.5 m and the radius of the base is 12 m?

Answer: 471.43 m²
We have to find the square metres of canvas required for the given conical tent.
Area of canvas required = Curved surface area of conical text.
We know that,
Curved surface area of cone=πrr2+h2
Where r is the radius of its base and height is h.
As given in the question, dimensions of conical tent are:
Height = 3.5 m
Radius = 12 m

On putting the values,
Curved surface area =227×12(12)2+(3.5)2
=227×12×156.25
=227×12×12.5
=471.43 m²
So, the area of the canvas =471.43 m2
Therefore, the area of canvas required is 471.43 m2.

Question:5

Two solid spheres made of the same metal have weights of 5920 g and 740 g, respectively. Determine the radius of the larger sphere if the diameter of the smaller one is 5 cm.

Answer: 5 cm
The density (D) of an object is mass (m) per unit volume (v).

D=mv
So mass is directly proportional to volume for the same metal (as density remains the same)
For Solid 1
Let Mass = M1
Volume = V1
For solid 2
Mass = M2
Volume = V2
Now, M1M2=V1V2
In the question, the objects are spheres
Volume of sphere =43πR3 (where R is radius)
So, volume is directly proportional to R3
Hence, M1M2=V1V2=R13R23
Putting the given values (5920g and 740g)
⇒ 5920740=R13R23
⇒ R13R23=8
⇒ R1R2=83
⇒ R1R2=2
So, R1=R2×2
The diameter of the smaller one is 5 cm
So, R2=52=2.5cm
R1=2.5×2
Hence, R1=5cm
Therefore radius of the larger sphere is 5 cm.

Question:6

A school provides milk to the students daily in cylindrical glasses of diameter 7 cm. If the glass is filled with milk up to an height of 12 cm, find how many litres of milk is needed to serve 1600 students.

Answer: 738.528 litres of milk
Given that a school provides milk to the students daily in cylindrical glasses of diameter 7 cm
So glass is a cylinder with a diameter = 7 cm
Radius of cylinder (r)=d2
= 7/2 = 3.5cm
Now milk is filled in this cylindrical glass.
Height of milk up to which the glass is filled (h)=12cm
So, the volume of milk filled in the cylindrical glass
V=πr2h (r is the radius, h is the height up to which the milk is filled)
Putting the values,
V = 227×3.5×3.5×12
⇒ 461.58cm3
Now, the number of students=1600=Total number of glasses required
Volume of 1600 cylindrical glasses = 1600 (Volume of one glass)
= 461.58×1600
= 738528cm3
Now we know that 1 litre = 1000 cm³
Volume of 1600 cylindrical glasses = 7385281000
=738.528litre
Hence, to serve 1600 students, we need 738.528 litres of milk.

Question:7

A cylindrical roller 2.5m in length, 1.75m in radius, when rolled on a road, was found to cover the area of 5500 m2. How many revolutions did it make?

Answer: 200 revolutions
It is given that when the cylindrical roller is rolled on the road, its covers 5500 m2 area
Now,
Area covered by cylindrical roller in one revolution = curved surface area of cylinder
And we know that,
Curved surface area of cylinder =2πrh (where r is the radius and h is its height)
Given length of roller =2.5m (This will be taken as height of the roller)
Cylindrical roller's radius = 1.75 m
So, Curved surface area =2×3.14×1.75×2.5
= 6.28×4.375 m2
= 27.475 m2
Total Area covered in one revolution = 27.475 m2
Now total area covered = 5500 m2
So, the Number of revolutions = the Total area covered in one revolution
= 5500/27.475
= 200.181≅200
Hence, the cylindrical roller makes 200 revolutions.

Question:8

A small village, having a population of 5000, requires 75 litres of water per head per day. The village has got an overhead tank of measurement 40m×25m×15m. For how many days will the water of this tank last?

Answer: 40 days
Given, dimension of overhead tank =40m×25m×15m
Now the tank is in the shape of a cuboid, so we have volume =lbh
⇒ Volume of tank =40m×25m×15m=15000m3
Population of village =5000
Daily water requirement for 1 person = 75 litre
Then daily water requirement for 5000 people =75×5000=375000l
Now we know that 1 litre =0.001m3
So, daily water requirement for 5000 people =375000×0.001m3=375m3
Number of Days till the water in this tank will last
=Volume of tankdaily water requirement for 5000 people
=15000m3375m3=40
Hence, the water of this tank will last for 40 days.

Question:9

A shopkeeper has one spherical laddoo of radius 5cm. With the same amount of material, how many laddoos of radius 2.5 cm can be made?

Answer: 8 laddoos
Given that a shopkeeper has:
Larger laddoo with radius = 5 cm
Smaller laddoo with radius = 2.5 cm
Volume of sphere =43πR3 (where R is the radius)
Volume of 1 larger laddoo =43π(5)3
Volume of 1 smaller laddoo =43π(2.5)3
Volume of 1 larger laddoo = Total volume of all smaller laddoos formed
Let number of smaller laddoos be n.
Volume of n smaller laddoo =n(43π(2.5)3)
So,
⇒43π×53=n×43π×(2.5)3
⇒125=n×(2510)3
⇒n=8laddoos
Hence 8 smaller laddoos can be formed by the larger laddoo.

Question:10

A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the volume and the curved surface of the solid so formed.

Answer: Volume =301.44cm3 and Curved surface area =188.57cm2
A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm.
So on revolving we get a right circular cone as shown in the figure.

Radius = 6 cm, Height = 8 cm, Slant Height = 10 cm
Now, we know that Volume of cone=13πr2h (where r is radius and h is height)
V=13×3.14×(6)2×8
=13×3.14×36×8
=13×12×8
=301.44cm3
Curved surface area =πrl (where r is radius and l is slant height)
=227×6×10
=13207
=188.57cm²
Hence, Volume = 301.44 cm³ and curved surface area =188.57 cm².

Question:1

A cylindrical tube opened at both the ends is made of iron sheet which is 2 cm thick. If the outer diameter is 16 cm and its length is 100 cm, find how many cubic centimeters of iron has been used in making the tube?

Answer: 8792 cm³
We know that
Volume of cylinder is given as πr2h (where r = outer radius and h is the height)
Here, the length of the cylinder will be considered as its height
(Height=length=100 cm)
Outer diameter is given as, d=16cm
So, outer radius r=162=8cm
Thus volume of cylinder =πr2h
=3.14×(8)2×100
=20096cm3
Thickness of iron sheet = 2 cm
Now, inner diameter = outer diameter – 2 (Thickness of iron sheet)
Inner diameter =16−(2×2)=12cm
So, inner radius (R)=122=6cm
Now, Volume of inner cylinder =πr2h, (where R = inner radius)
=πR2h
=3.14×(6)2×100
=11304cm3
Thus, Volume of iron used = Volume of outer cylinder - Volume of inner cylinder
=(20096−11304)cm3
=8792cm3
Hence, the answer is 8792 cm3.

Question:2

A semi-circular sheet of metal of diameter 28cm is bent to form an open conical cup. Find the capacity of the cup.

Answer: 622.38 cm³
It is given that a semi-circular sheet is bent to form an open conical cup.
Now, the radius of the sheet becomes the slant height of the cup and the circumference of the sheet becomes the circumference of the base of the cone.
Diameter of semi-circular sheet =28cm
So, radius of semi-circular sheet (R)=14cm
Slant height of the conical cup (l)=14cm
Now,
Let the radius of base of the cup = r cm.
Circumference of semi-circular sheet = Circumference of base of conical cup
πR=2πr
⇒227×14=2.227.r
⇒r=7cm
Let the height of cup be h cm.
Then we know that for a right circular cone,
l2=r2+h2
⇒(14)2=(7)2+h2
⇒196−49=h2
⇒h=73cm
∴ Capacity of the cup = Volume of a cone =13πr2h
=13.227.(7)2.73
=107833
=10783×1.732
=622.38cm3

Question:3

A cloth having an area of 165 m2 is shaped into the form of a conical tent of radius 5m

1. How many students can sit in the tent if a student, on average, occupies 57 m2 on the ground?
2. Find the volume of the cone.

Answer: (i) 110 students
(ii) 241.73 m3
We have, Area of cloth = 165 m2
This cloth is shaped in the form of a conical tent.
Radius of conical tent = 5 cm
So, we have:
Area of cloth = Curved surface area of cone
Curved surface area of cone is given as πrl Where, r = radius of a cone
l = slant height of a cone
So curved surface area of conical tent,
165=227×5×l
⇒l=165×722×5=212=10.5m
(i) Now it is given that area covered by 1 student =57m2
So the number of students=Area of a circular base of a coneArea covered by 1 student
⇒Number of student=πr257=(227×52)57=110
So 110 students can sit in the tent.
(ii) Volume of cone : 13πr2h
For a right circular cone, we have
r2+h2=l2
⇒52+h2=(10.5)2
⇒25+h2=110.25
⇒h2=110.25−25=85.25
h=85.25=9.23m
So, volume =13×227×52×9.23
=5076.521=241.73m3

Question:4

The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Calculate the volume of water pumped into the tank.

Answer: 668.66 m3
We know that Volume of hemispherical tank=23πr3
Where, r = radius of hemispherical tank
Now, internal diameter of hemispherical tank is given as 14 m
So, internal radius of hemispherical tank=142=7m
Tank contains 50 kL of water
We have, 1000L=1m3
So, 1kL=1m3
This means that tank contains 50m3 of water
⇒ Volume of hemispherical tank =23×227×(7)3=21563=718.66m3
Now, as we are given that the already tank contains 50 kilolitres of water
Volume of water pumped into the tank = Total Volume of hemispherical tank −50m3
Volume of water pumped into the tank =(718.66−50)m3=668.66m3
Hence, the answer is 668.66m3

Question:5

The volumes of the two spheres are in the ratio 64:27. Find the ratio of their surface areas.

Answer: 16:9
We have been given that the volume of two sphere is in the ratio =64:27
We know that, Volume of sphere is given as 43πr3 (where r is its radius)
And surface area is given as 4πr2
Let volume of sphere 1=V1 (radius r1)
And, volume of sphere 2=V2 (radius r2)
Then, V1V2=6427
43πr1343πr23=6427
r13r23=(43)3
r1r2=43
Then, ratio of areas of both spheres
area of sphere (1))area of sphere (2)=4πr124πr22
=r12r22=(r1r2)2=(43)2=169
Hence the required ratio is 16:9

Question:6

A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.

Answer: 30.47cm3
Volume of cube is given as (side length)3=(4cm)3=64cm3
As the cube contains the sphere,
Diameter of sphere = side length of the cube = 4 cm
Radius of sphere (r)=42cm=2cm
We know that volume of a sphere is given as 43πr3 (where r is the radius)
Volume =43.227.2.2.2=70421cm3
∴Volume of the gap in between=Volume of cube−volume of sphere
=(64−70421)cm3
=64021cm3
=30.47cm3
Hence the required answer is 30.47cm3

Question:7

A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?

Answer: 50%
Let the radius of the sphere be r, radius of cylinder be R, height of cylinder be H
Volume of sphere (43πr3) = Volume of cylinder (πR2H)
Now, it is given that r = R
(43πr3)=(πr2H)
43r=h
4r=3h
2(2r)=3h (∴d=2r)
⇒2d=3h
d=1.5h=(1+0.5)h
d=(1+50100)h
The diameter exceeds the height by 50%.

Question: 8(i)

(i) 30 circular plates, each of radius 14 cm and thickness 3cm are placed one above the another to form a cylindrical solid. Find: the total surface area
(ii) 30 circular plates, each of radius 14 cm and thickness 3 cm are placed one above the another to form a cylindrical solid. Find the volume of the cylinder so formed.

(i) Answer: 9152 cm²
It is given that 30 circular plates are placed one above the other to form a cylindrical solid.
So we can see that the height of this cylindrical solid = the total thickness of 30 circular plates placed one over the other
Given, the thickness of one circular plate =3cm
Thickness of 30 circular plates =30×3=90cm
Now, the radius of the circular plate, r = 14 cm
We know that
The total surface area of the cylindrical solid of formed will be equal to 2πr(h+r)
Where r is its radius and h is the height. So, putting the above calculated values:
Total surface area =2×227×14(90+14)
=44×2×104
=9152 cm²
Hence, the total surface area of the cylindrical solid is 9152 cm².
(ii)
Answer: 55440 cm³
It is given that 30 circular plates are placed one above the other to form a cylindrical solid.
So we can see that the height of this cylindrical solid = the total thickness of 30 circular plates placed one over the other
Given, the thickness of one circular plate = 3 cm
Thickness of 30 circular plates = 30×3 = 90cm
Now, the radius of the circular plate, r = 14 cm
We know that
Volume of the cylinder so formed will be equal to πr²h
Where r is its radius and h is the height. So, putting the above calculated values:
Volume of the cylinder = 22/7 × (14)² × 90
= 22/7 × 14 × 14 × 90
= 55440 cm³
Hence, the volume of the cylindrical solid is 55440cm³.