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RD Sharma Class 10 Solutions Chapter 14 Surface Areas and Volumes Ex 14.1
How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius 8 cm ?
How many spherical bullets each of 5 cm in diameter can be cast from a rectangular block of metal 11dm x 1m x 5 dm?
Volume of reactangular block of metal
(V1) = 11 dm x 1 m x 5 dm
= 11 dm x 10 dm x 5 dm = 550 dm3
Diameter of spherical bullet = 5 cm
A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of the two balls are 1.5 cm and 2 cm respectively. Determine the diameter of the third ball.
2.2 cubic dm of brass is to be drawn into a cylindrical wire 0.25 cm in diameter. Find the length of the wire.
What length of a solid cylinder 2 cm in diameter must be taken to recast into a hollow cylinder of length 16 cm, external diameter 20 cm and thickness 2.5 mm ?
Length of hollow cylinder = 16 cm
External diameter = 20 cm
and thickness = 2.5 mm = 0.25 cm
∴External radius (R) = 202 = 10 cm
and internal radius (r) = 10 – 0.25 = 9.75 cm
∴ Volume of the material used in hollow cylinder
= πh (R² – r²) = π x 16 (10² – 9.75²) cm²
= 16π (100.00 – 95.0625) = 16TC (4.9375) cm3
∴ Volume of solid cylinder = 16 x 4.93 75πcm3 Diameter = 2 cm
∴Radius (r) = 22 = 1 cm
Let h be the height of the cylinder, then
πr2h = 16 x 4.9375π
⇒ π (1)2 h = 16 x 4.93 75π
h = 16 x 4.9375 = 79
Hence height = 79 cm
A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter 42 cm and jteight 21 cm which are filled completely. Find the diameter of the cylindrical vessel.
Diameter of each small cylindrical vessels = 42 cm
∴Radius of each vessel (r) = 422 = 21 cm
Height (h) = 21 cm
∴ Volume of each cylindrical vessal = πr2h = π (21 )2 (21) = 9261K cm3
and volume of both vessels = 2 x 9261π = 18522π cm3
Now volume of larger cylindrical vessel = 185 22π cm3
Let R be the radius of the vessel, then Height (H) = diameter = 2R,
∴Volume πR2H = πR2 x 2R = 2πR2
∴ 2πR3 = 18522π
⇒ R3 = 185222 = 9261
⇒ R3 = 9261 = (21 )3
∴ R = 21
∴Diameter = 2R = 2×21=42 cm
50 circular plates each of diameter 14 cm and thickness 0.5 cm are placed one above the other to form a right circular cylinder. Find its total surface area.
Diameter of each circular plate = 14 cm
∴ Radius (r) = 142 = 7 cm .
Thickness (h) = 0.5 cm
∴Height of 50 plates placing one above the other
= 0.5 x 50 = 25 cm
∴ Curved surface area of the cylinder so formed = 2πrh
= 2 x 27 x 7 x 25 = 1100 cm2
and total surface area = curved surface area + 2 x base area
= 1100 + 2 x πr2 =1100+ 2 x 222 x 7 x 7
= 1100 + 308 = 1408 cm2
25 circular plates, each of radius 10.5 cm and thickness 1.6 cm, are placed one above the other to form a solid circular cylinder. Find the curved surface area and the volume of the cylinder so formed.
Find the number of metallic circular discs with 1.5 cm base diameter and of height 0. 2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm. [NCERT Exemplar]
Given that, lots of metallic circular disc to be melted to form a right circular cylinder. Here, a circular disc work as a circular cylinder.
Base diameter of metallic circular disc = 1.5 cm
∴Radius of metallic circular disc = 1.52 cm [∵ diameter = 2 x radius]
and height of metallic circular disc i.e., = 0.2 cm
∴ Volume of a circular disc = π x (Radius)2 x Height
How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimensions 66 cm x 42 cm x 21 cm. [NCERT Exemplar]
How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm. [NCERT Exemplar]
Hence, the required number of spherical lead shots is 2541.
Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is 12 3–√ cm. Find the edges of the three cubes. [NCERT Exemplar]
Let the edges of three cubes (in cm) be 3x, 4x and 5x, respectively.
Volume of the cubes after melting is = (3x)3 + (4x)3 + (5x)3 = 216×3 cm3
Let a be the side of new cube so formed after melting. Therefore, a3 = 216x3
So, a = 6x,
A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. [NCERT Exemplar]
The volume of the solid metallic sphere = 43 π(10.5)3 cm3
Volume of a cone of radius 3.5 cm and height 3 cm = 13 π (3.5)2 x 3 cm3
Number of cones so formed
The diameter of a metallic sphere is equal to 9 cm. It is melted and drawn into a long wire of diameter 2 mm having uniform cross-section. Find the length of the wire.
An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is 14 of the radius of the original ball, how many such balls are made ? Compare the surface area, of all the smaller balls combined together with that of the original ball.
Let R be the radius of the original ball, then
A copper sphere of radius 3 cm is melted and recast into a right circular cone of height 3 cm. Find the radius of the base of the cone.
A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.
The diameters of internal and external surfaces of a hollow spherical shell are 10 cm and 6 cm respectively. If it is melted and recast into a solid cylinder of length of 2 23 cm, find the diameter of the cylinder.
How many coins 1.75 cm in diameter and 2 mm thick must be melted to form a cuboid 11 cm x 10 cm x 7 cm ?
The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into a cone of height 28 cm. Find the diameter of the base of the cone so formed. (Use π = 22/7). [CBSE 2010]
A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied out on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. [CBSE 2012]
A solid metallic sphere of radius 5.6 cm is melted and solid cones each of radius 2.8 cm and height 3.2 cm are made. Find the number of such cones formed. [CBSE 2014]
A solid cuboid of iron with dimensions 53 cm x 40 cm x 15 cm is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are 8 cm and 7 cm respectively. Find the length of pipe. [CBSE 2015]
Length of cuboid (l) = 53 cm
Breadth (b) = 40 cm
Height (h) = 15 cm
Volume =lbh= 53 x 40 x 15 cm3
= 31800 cm3
∴ Volume of cylindrical pipe = 31800 cm3
Inner diameter of pipe = 7 cm
and outer diameter = 8 cm
∴ Outer radius (R) = 82 = 4 cm
The diameters of the internal and external surfaces of a holjow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder. [CBSE 2001C]
Outer diameter of hollow spherical shell = 10 cm
and inner diameter = 6 cm
Outer radius (R) = 102 = 5 cm 6
and inner radius (r) = 62 = 3 cm
A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Calculate the height of the cone.
A hollow sphere of internal and external radii 2 cm and 4 cm respectively is melted into a cone of base radius 4 cm. Find the height and slant height of the cone.
Interval radius of hollow sphere (r) = 2m
and external radius (R) = 4m
A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of the balls are 1.5 cm and 2 cm. Find the diameter of the third ball.
Radius of big spherical ball (R) = 3 cm
∴ Volume = 43 πR3 = 43 n x (3)3 cm3
= 43 π x 27 = 36π cm3
Similarly volume of ball of radius (r1) = 1.5 cm
∴ Volume = 43 π( 1.5)3
A path 2 m wide surrounds a circular pond of diameter 40 m. How many cubic metres of gravel are required to grave the path to a depth of 20 cm ?
Diamter of pond = 40 m
∴ Radius (r) = 402 =20 m
Width of path = 2 m
A 16 m deep well with diameter 3.5 m is dug up and the earth from it is spread evenly to form a platform 27.5 m by 7 m. Find the height of the platform.
Volume of platform = Ibh = 27.5 x7 x h
A well of diameter 2 m is dug 14 m deep. The earth taken out of it is spread evenly all around it to form an embankment of height 40 cm. Find the width of the embankment.
Diameter of well = 2 m
A well with inner radius 4 m is dug 14 m deep. Earth taken out of it has been spread evenly all around a width of 3 m it to form an embankment. Find the height of the embankment.
A well of diameter 3 m is dug 14 m deep. The earth taken out of it has>been spread evenly all around it to a width of 4 m to form an embankment. Find the height of the embankment.
Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 9 cm.
A cylindrical bucket, 32 cm high and 18 cm of radius of the base, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.
Rain water, which falls on a flat rectangular surface of length 6 m and breadth 4 m is transferred into a cylindrical vessel of internal radius 20 cm. What will be the height of water in the cylindrical vessel if a rainfall of 1 cm has fallen ? [Use π = 22/7]
The rain water from a roof of dimensions 22 m x 20 m drains into a cylindrical vessel having diameter of base 2 m and height 3.5 m. If the rain water collected from the roof just fills the cylindrical vessel, then find the rain in cm. [NCERT Exemplar]
A conical flask is full of water. The flask has base-radius r and height h. The water is poured into a cylindrical flafisk of base- radius mr. Find the height of water in the cylindrical flask.
A rectangular tank 15 m long and 11 m broad is required to receive entire liquid contents from a full cylindrical tank of internal diameter 21m and length 5 m. Find the least height of the tank that will serve the purpose.
A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be fdled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are necessary to empty the bowl ?
A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical ball is dropped into the tub and the level of the water is raised by 6.75 cm. Find the radius of the ball.
500 persons have to dip in a rectangular tank which is 80 m long and 50 m broad. What is the rise in the level of water inthe tank, if the average displacement of water by a person is 0.04 m3 ?
A cylindrical jar of radius 6 cm contains oil. Iron spheres each of radius 1.5 cm are immersed in the oil. How many spheres are necessary to raise the level of the oil by two centimetres ?
A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical form ball of radius 9 cm is dropped into the tub and thus the level of water is raised by h cm. What is the value of h ?
Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimension 16 cm x 8 cm x 8 cm when 16 spheres are packed the box is fdled with preservative liquid. Find the volume of this liquid. [Use π = 669/213]
A vessel in the shape of a cuboid contains some water. If three identical spheres are immersed in the water, the level of water is increased by 2 cm. If the area of the base of the cuboid is 160 cm² and its height 12 cm, determine the radius of any of the spheres.
150 spherical marbles, each of diameter 1.4 cm are dropped in a cylindrical vessel of diameter 7 cm contianing some water, which are completely immersed in water. Find the rise in the level of water in the vessel. [CBSE2014]
Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which (25)² of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant? [CBSE 2014]
Radius of vessel (R) = 2.5 cm
and height (h) = 11 cm
16 glass spheres each of radius 2 cm are packed into a cuboidal box of internal dimensions 16 cm x 8 cm x 8 cm and then the box is filled with water. Find the volume of water fdled in the box. |NCERT Exemplar]
Given, dimensions of the cuboidal
= 16 cm x 8 cm x 8 cm
∴ Volume of the cuboidal =16x8x8
= 1024 cm3
Also, given radius of one glass sphere = 2cm
∴ Volume of one glass sphere = (43)πr³
= (43) x (227) x (2)³
= (70421) = 33.523 cm³
Now, volume of 16 glass spheres = 16 x 33.523 = 536.37 cm3
∴ Required volume of water = Volume of cuboidal – Volume of 16 glass spheres
= 1024 – 536.37 = 487.6 cm3
Water flows through a cylindrical pipe, whose inner radius is 1 cm, at the rate of 80 cm/sec in an empty cylindrical tank, the radius of whose base is 40 cm. What is the rise of water level in tank in half an hour? [NCERT Exemplar]
Given, radius of tank, r1 = 40 cm
Let height of water level in tank in half an hour = h1
Also, given internal radius of cylindrical pipe, r1 = 1 cm
and speed of water = 80 cm/s /., in 1 water flow = 80 cm
∴In 30 (min) water flow = 80 x 60 x 30 = 144000 cm
According to the question,
Volume of water in cylindrical tank = Volume of water flow the circular pipe in half an hour
⇒ πr1²h1 = πr2²h2
⇒ 40 x 40 x h1 = 1 x 1 x 14400
∴ h1 = (14400040×40) = 90 cm
Hence, the level of water in cylindrical tank rises 90 cm in half an hour.
Water in a canal 1.5 m wide and 6 m deep is flowing with a speed of 10km/hr. How much area will it irrigate in 30 minutes if 8 cm of standing water is desired ?
A famer runs a pipe of internal diameter 20 cm from the canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled ?
A cylindrical tank full of water is emptied by a pipe at the rate of 225 litres per minute. How much time will it take to empty half the tank, if the diameter of its base is 3 m and its height is 3.5 m? [Use π = 22/7] [ [CBSE 2014]
A cylindrical tank full of water is emptied by a pipe at the rate of 225 litres per minute. How much time will it take to empty half the tank, if the diameter of its base is 3 m and its height is 3.5 m? [Use π = 22/7] [CBSE 2014]
Water flows at the rate of 15 km/hr through a pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?
Given, length of the pond = 50 m and width of the pond = 44 m
A canal is 300 cm wide and 120 cm deep. The water in the canal is flowing with a speed of 20 km/h. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired? [NCERT Exemplar]
Volume of water flows in the canal in one hour = width of the canal x depth of the canal x speed of the canal water
= 3 x 1.2 x 20 x 1000 m³ = 72000 m³
In 20 minutes the volume of water
= 72000×2060 m³ = 24000 m³
Area irrigated in 20 minutes, if 8 cm,
i. e., 0.08 m standing water is required
= (25) m²
= 300000 m²= 30 hectares.
The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 cm2, find the volume of cylinder. (Use re = 22/7) [CBSE 2016]
Let the radius and height of cylinder be r and h respectively.
r + h = 37 cm …(i) [given]
Total surface area of cylinder = 1628 cm²
2πr(r + h) = 1628
⇒ 2πr(37) = 1628
A tent of height 77 dm is in the form a right circular cylinder of diameter 36 m and height 44 dm surmounted by a right circular cone. Find the cost of the canvas at ₹3.50 per m². [Use π = 22/7]
Total height of the tent = 77 dm
Height of the cylindrical part (h1) = 44 dm = 4.4 m
Height of conical part (h2) = 77 – 44 = 33 dm = 3.3 m
Diameter of the base of the tent = 36 m
The largest sphere is to be curved out of a right circular cylinder of radius 7 cm and height 14 cm. Find the volume of the sphere.
Largest sphere to be curved must have its diameter = diameter of cylinder = height of cylinder
A right angled triangle whose sides are 3 cm, 4 cm and 5 cm is revolved about the sides containing the right angle in two ways. Find the difference in volumes of the two cones so formed. Also, find their curved surfaces.
In ∆ABC, ∠B = 90°
and sides are 3 cm, 4 cm and 5 cm
Here diagonal is 5 cm
(i) When it is revolved along BC i.e., 4 cm side, then
Radius of the base of the cone (r1) = 3 cm
and height (h1) = 4 cm and slant height l = 5 cm
∴ Volume = (25) πr2h = (13) π x 3 x 3 x 4 cm³
= 12π cm³
and surface area = πrl = π x 3 x 5 cm2 = 157π cm2
(ii) When it is revolved along 3 cm side, then
r = 4 cm and h = 3 cm and l = 5 cm
∴ Volume = (13) πr2h =-(13) π x 4 x 4 x 3 cm3
and curved surface area = πrl
= π x 4 x 5 = 20n cm²
∴ Difference of their volumes = I6π- 12π = 4πcm³
A 5 m wide cloth is used to make a conical tent of base diameter 14 ir. and height 24 m. Find the cost of cloth used at the rate of ₹25 per metre. [Use π = 22/7] [CBSE 2014]
Diameter of base of conical tent = 14 m
Radius (r) = (143)= 7 cm
Height (h) = 24 m
The volume of a hemi-sphere is 2425 (12) cm³, find its curved surface area. 
The difference between the outer and inner curved surface areas of a hollow right circular cylinder 14 cm long is 88 cm². If the volume of nretal used in making the cylinder is 176 cm³, find the outer and inner diameters of the cylinder. [CBSE 2010]
Height of hollow right cylinder = 14 cm
Difference between outer and inner curved surface area = 88 cm²
and volume of metal used =176 cm³
The internal and external diameters of a hollow hemispherical vessel are 21 cm and 25.2 cm respectively. The cost of painting 1 cm² of the surface is 10 paise. Find the total cost to paint the vessel ail over.
Outer diameter of the hemispherical vessel = 25.2 cm
and inner diameter = 21 cm
Prove that the surface area of a sphere is equal to the curved surface area of the circumscribed cylinder.
Let r be the radius of the sphere, then surface area = 4πr² ….(i)
Then radius of the circumscribed cylinder = radius of the sphere = r
If the total surface area of a solid hemisphere is 462 cm², find its volume (Take π = 22/7).
Water flows at the rate of 10 m / minutes through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm? [NCERT Exemplar]
Given, speed of water flow = 10 m min-1 = 1000 cm/min
A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius of the cone. [NCERT Exemplar]
(i) Whenever we placed a solid right circular cone in a right circular cylinder with full of water, then volume of a solid right circular cone is equal to the volume of water failed from the cylinder.
(ii) Total volume of water in a cylinder is equal to the volume of the cylinder.
(iii) Volume of water left in the cylinder = Volume of the right circular cylinder – Volume of a right circular cone.
Now, given that
Height of a right circular cone = 120 cm
Radius of a right circular cone = 60 cm
∴Volume of a right circular cone = (13)πr² x h
= (13) x (227) x 60 x 60 x 120
=(227) x 20 x 60 x 120
= 144000π cm³
∴Volume of a right circular cone = Volume of water failed from the cylinder = 1440007c cm3 [from point (i)]
Given that, height of a right circular cylinder = 180 cm
and radius of a right circular cylinder = Radius of a right circular cone = 60 cm
∴Volume of a right circular cylinder = πr² x h
= π x 60 x 60 x 180 = 64800071 cm3 So, volume of a right circular cylinder = Total volume of water in a cylinder = 64800071 cm3 [from point (ii)]
From point (iii),
Volume of water left in the cylinder = Total volume of water in a cylinder – Volume of water failed from the cylinder when solid cone is placed in it
= 648000π- 144000π
= 504000π = 504000 x (227) = 1584000 cm³ 1584000
= (1584000(10)6) m³ = 1.584 m³
Hence, the required volume of water left in the cylinder is 1.584 m³.
A heap of rice in the form of a cone of diameter 9 m and height 3.5 m. Find the volume of rice. How much canvas cloth is required to cover the heap? [NCERT Exemplar]
Given that, a heap of rice is in the form of a cone.
Height of a heap of rice i.e., cone (h) = 3.5m
and diameter of a heap of rice i. e., cone = 9 m
Radius of a heap of rice i.e., cone (r) = (92) m
A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the concial heap is 24 cm, find the radius and slant height of the’ heap. [NCERT Exemplar]
Given, radius of the base of the bucket = 18 cm
Height of the bucket = 32 cm
So, volume of the sand in cylindrical bucket
= πr²h = π (18)²x 32= 10368π
Also, given height of the conical heap (h) = 24 cm
Let radius of heap be r cm
Then, volume of the sand in the heap = (13) πr²h
A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed to empty the bowl? [NCERT Exemplar]
Given, radius of hemispherical bowl, r = 9cm
and radius of cylindrical bottles, R = 1.5 cm
and height, h = 4 cm
∴Number of required cylindrical bottles
A factory manufactures 120,000 pencils daily. The pencils are cylindrical in shape each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at ₹0.05 per dm². [NCERT Exemplar]
Given, pencils are cylindrical in shape.
Length of one pencil = 25 cm
and circumference of base = 1.5 cm
Now, cost of colouring 1 dm2 curved surface of the pencils manufactured in one day = ₹0.05
∴ Cost of colouring 45000 dm2 curved surface = ₹2250
The(34)th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.
Let height of water in cylindrical vessel = h cm
Volume of water (in cylinder) = (34) Volume of water (in come)
∵ Volume of cylinder = nr²h & Volume of cone = (13) πr²h
Hence, height of water in cylindrical vessel (h) = 1.5 cm
All Subject NCERT Exemplar Problems Solutions For Class10
All Subject NCERT Solutions For Class 10
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