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RD Sharma solutions for Class 10 Mathematics chapter 14 - Surface Areas and Volumes

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Chapter 14: Surface Areas and Volumes

Ex. 14.10Ex. 14.20Ex. 14.30Others

Chapter 14: Surface Areas and Volumes Exercise 14.10 solutions [Pages 27 - 32]

Ex. 14.10 | Q 1 | Page 27

How many balls each of radius 1cm can be made from a solid sphere of lead of radius
8cm?

Ex. 14.10 | Q 2 | Page 27

How many spherical bullets each of 5cm in diameter can be cast from a rectangular block of metal 11 dm x 1m x 5 dm?

 

Ex. 14.10 | Q 3 | Page 27

A spherical ball of radius 3cm is melted and recast into three spherical balls. The radii of the two of balls are 1.5cm and 2cm . Determine the diameter of the third ball?

Ex. 14.10 | Q 4 | Page 27

2.2 Cubic dm of grass is to be drawn into a cylinder wire 0.25cm in diameter. Find the length of wire?

Ex. 14.10 | Q 5 | Page 28

What length of a solid cylinder 2cm in diameter must be taken to recast into a hollow
cylinder of length 16cm, external diameter 20cm and thickness 2.5mm?

Ex. 14.10 | Q 6 | Page 28

A cylindrical vessel having diameter equal to its height is full of water which is poured into two identical cylindrical vessels with diameter 42cm and height 21cm which are filled completely. Find the diameter of cylindrical vessel?

Ex. 14.10 | Q 7 | Page 28

50 circular plates each of diameter 14cm and thickness 0.5cm  are placed one above other to form a right circular cylinder. Find its total surface area?

Ex. 14.10 | Q 8 | Page 28

25 circular plates each of radius 10.5cm  and thickness 1.6cm are placed one above the other to form a solid circular cylinder. Find the curved surface area and volume of cylinder so formed?

Ex. 14.10 | Q 9 | Page 28

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 .

Ex. 14.10 | Q 10 | Page 28

How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimension  6cm \[\times\] 42cm \[\times\] 21 cm.

Ex. 14.10 | Q 11 | Page 28

How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm . 

Ex. 14.10 | Q 12 | Page 28

Three cubes  of a metal whose edges are in the ratios 3 : 4 : 5 are melted and converted into a single cube whose diagonal is  \[12\sqrt{3}\]. Find the edges of the three cubes.

Ex. 14.10 | Q 13 | Page 28

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.

 
Ex. 14.10 | Q 14 | Page 28

The diameter of a metallic sphere is equal to 9cm. it is melted and drawn into a long wire of diameter 2mm having uniform cross-section. Find the length of the wire?

Ex. 14.10 | Q 15 | Page 28

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 1/4 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.

Ex. 14.10 | Q 16 | Page 28

A copper sphere of radius 3cm is melted and recast into a right circular cone of height 3cm.find radius of base of cone?

Ex. 14.10 | Q 17 | Page 28

A copper rod of diameter 1cm and length 8cm is drawn into a wire of length 18m of uniform thickness. Find thickness of wire?

Ex. 14.10 | Q 18 | Page 28

The diameters of internal and external surfaces of hollow spherical shell are 10cm and 6cm respectively. If it is melted and recast into a solid cylinder of length of 2`2/3`cm, find the
diameter of the cylinder.

Ex. 14.10 | Q 19 | Page 28

How many coins 1.75cm  in diameter and 2mm thick must be melted to form a cuboid 11cm x 10cm x 75cm___?

Ex. 14.10 | Q 20 | Page 28

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 it =`22/7`)

Ex. 14.10 | Q 21 | Page 29

A cylindrical bucket, 32 cm high and with radius of base 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 conical heap is 24 cm. Find the radius and slant height of the heap.

Ex. 14.10 | Q 22 | Page 29

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. 

Ex. 14.10 | Q 23 | Page 29

A solid cuboid of iron with dimensions 53 cm ⨯ 40 cm ⨯ 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.

Ex. 14.10 | Q 24 | Page 29

The diameters of the internal and external surfaces of a hollow spherical shell are 6 cm and 10 cm respectively. If it is melted and recast and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder.

Ex. 14.10 | Q 25 | Page 29

A hollow sphere of internal and external diameter 4cm and 8cm is melted into a cone of base diameter 8cm. Calculate height of cone?

Ex. 14.10 | Q 26 | Page 29

A hollow sphere of internal and external radii 2cm and 4cm is melted into a cone of basse radius 4cm. find the height and slant height of the cone______?

Ex. 14.10 | Q 27 | Page 29

A spherical ball of radius 3cm is melted and recast into three spherical balls. The radii of the two of balls are 1.5cm and 2cm . Determine the diameter of the third ball?

Ex. 14.10 | Q 28 | Page 29

A path 2m wide surrounds a circular pond of diameter 40m. how many cubic meters of gravel are required to grave the path to a depth of 20cm ?

Ex. 14.10 | Q 29 | Page 29

A 16m deep well with diameter 3.5m is dug up and the earth from it is spread evenly to form a platform 27.5m by 7m. Find height of platform?

Ex. 14.10 | Q 30 | Page 29

A well of diameter 2m is dug14m deep. The earth taken out of it is spread evenly all around it to form an embankment of height 40cm. Find width of the embankment?

Ex. 14.10 | Q 31 | Page 29

A well with inner radius 4m is dug 14m deep earth taken out of it has been spread evenly all around a width of 3m it to form an embankment. Find the height of the embankment?

Ex. 14.10 | Q 32 | Page 29

A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

Ex. 14.10 | Q 33 | Page 29

Find the volume of the largest right circular cone that can be cut out of a cube where edgeis 9cm?

Ex. 14.10 | Q 34 | Page 29

A cylindrical bucket, 32 cm high and with radius of base 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 conical heap is 24 cm. Find the radius and slant height of the heap.

Ex. 14.10 | Q 35 | Page 29

Rain water, which falls on a flat rectangular surface of length 6cm and breath 4m is
transferred into a cylindrical vessel of internal radius 20cm. What will be the height of
water in the cylindrical vessel if a rainfall of 1cm has fallen____?

Ex. 14.10 | Q 36 | Page 29

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3·5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

Ex. 14.10 | Q 37 | Page 29

A conical flask is full of water. The flask has base radius and height h. The water is poured into a cylindrical flask of base-radius mr. Find the height of water in the cylindrical flask.

Ex. 14.10 | Q 38 | Page 30

A rectangular tank 15m long and 11m broad is required to receive entire liquid contents from a full cylindrical tank of internal diameter 21m and length 5m. Find least height of tank that will serve purpose .

Ex. 14.10 | Q 39 | Page 30

A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are necessary to empty the bowl?

Ex. 14.10 | Q 40 | Page 30

A cylindrical tube of radius 12cm contains water to a depth of 20cm. A spherical ball is dropped into the tube and the level of the water is raised by 6.75cm.Find the radius of the ball .

Ex. 14.10 | Q 41 | Page 30

500 persons have to dip in a rectangular tank which is 80m long and 50m broad. What is the rise in the level of water in the tank, if the average displacement of water by a person is 0.04m3 .

Ex. 14.10 | Q 42 | Page 30

A cylindrical jar of radius 6cm contains oil. Iron sphere each of radius 1.5cm are immersed in the oil. How many spheres are necessary to raise level of the oil by two centimetress?

Ex. 14.10 | Q 43 | Page 30

A cylindrical tube of radius 12cm contains water to a depth of 20cm. A spherical ball of radius 9cm is dropped into the tube and thus level of water is raised by hcm. What is the value of h.

Ex. 14.10 | Q 44 | Page 30

Metal spheres each of radius 2cm are packed into a rectangular box of internal dimension 16cm x 8cm x 8cm when 16 spheres are packed the box is filled with preservative liquid. Find volume of this liquid?

Ex. 14.10 | Q 45 | Page 30

A vessel in the shape of cuboid ontains some water. If these identical spheres are immersed in the water, the level of water is increased by 2cm. if the area of base of cuboid is 160cm2 and its height 12cm, determine radius of any of spheres?

Ex. 14.10 | Q 46 | Page 30

150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in  water. Find the rise in the level of water in the vessel.

Ex. 14.10 | Q 47 | Page 30

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 2/th 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?

Ex. 14.10 | Q 48 | Page 30

16 glass spheres each of radius  2 cm are packed into a cuboidal box of internal dimensions  \[16 cm \times 8 cm \times 8 cm\] and then the box is filled with water . Find the volume of the water filled in the box .

Ex. 14.10 | Q 49 | Page 30

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 ? 

Ex. 14.10 | Q 50 | Page 30

Water in a canal 1.5m wide and 6m deep is flowering with a speed of 10km/ hr. how much area will it irrigate in 30 minutes if 8cm of standing water is desired?

Ex. 14.10 | Q 51 | Page 30

A farmer 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?

 
Ex. 14.10 | Q 52 | Page 31

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 \[\pi = \frac{22}{7}\]]

Ex. 14.10 | Q 53 | Page 31

Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe.

Ex. 14.10 | Q 54 | Page 31

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.  

Ex. 14.10 | Q 55 | Page 31

A canal is 300 cm wide and 120 cm deep. The water in the canal is flowing with a speed of 20 km/hr. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired ?

Ex. 14.10 | Q 56 | Page 31

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 sq. cm, find the volume of the cylinder. `("use " pi=22/7)`

Ex. 14.10 | Q 57 | Page 31

A tent of height 77dm is in the form a right circular cylinder of diameter 36m and height 44dm surmounted by a right circular cone. Find the cost of canvas at Rs.3.50 per m2 ?

Ex. 14.10 | Q 58 | Page 31

The largest sphere is to be curved out of a right circular of radius 7cm and height 14cm. find volume of sphere?

Ex. 14.10 | Q 59 | Page 31

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 days. Find the difference in columes of the two cones so formed. Also, find their curved surfaces.

Ex. 14.10 | Q 60 | Page 31

A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of Rs 25 per metre ?\[[Use \pi = \frac{22}{7}]\]

 

Ex. 14.10 | Q 61 | Page 31

The volume of a hemisphere is 2425`1/2cm^3`cm. Find its curved surface area?

Ex. 14.10 | Q 62 | Page 31

The difference between outer and inner curved surface areas of a hollow right circular cylinder 14cm long is 88cm2. If the volume of metal used in making cylinder is 176cm3.find the outer and inner diameters of the cylinder____?

Ex. 14.10 | Q 63 | Page 31

The internal and external diameters of a hollow hemisphere vessel are 21cm and 25.2 cm The cost of painting 1cmof the surface is 10paise. Find total cost to paint the vessel all
over______?

Ex. 14.10 | Q 64 | Page 31

Prove that the surface area of a sphere is equal to the curved surface area of the circumference cylinder__?

Ex. 14.10 | Q 65 | Page 31

If the total surface area of a solid hemisphere is 462 cm2 , find its volume.[Take π=22/7]

 

Ex. 14.10 | Q 66 | Page 31

Water flows at the rate of 10 m / minute  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.

Ex. 14.10 | Q 67 | Page 31

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 te cone 

Ex. 14.10 | Q 68 | Page 32

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  ?

Ex. 14.10 | Q 69 | Page 32

A cylindrical bucket, 32 cm high and with radius of base 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 conical heap is 24 cm. Find the radius and slant height of the heap.

Ex. 14.10 | Q 70 | Page 32

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 ?

Ex. 14.10 | Q 71 | Page 32

A factory manufactures 120,000 pencils daily . The pencil 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.

Ex. 14.10 | Q 72 | Page 32

The `3/4` 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. 

Chapter 14: Surface Areas and Volumes Exercise 14.20 solutions [Pages 60 - 63]

Ex. 14.20 | Q 1 | Page 60

A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of cylinder is 24 m. The height of the cylindrical portion is 11 m while the vertex of the cone is 16 m above the ground. Find the area of canvas required for the tent.

Ex. 14.20 | Q 2 | Page 60

A rocket is in the form of a circular cylinder closed at the lower end with a cone of the same radius attached to the top. The cylinder is of radius 2.5m and height 21m and the cone has a slant height 8m. Calculate total surface area and volume of the rocket?

Ex. 14.20 | Q 3 | Page 60

A tent of height 77dm is in the form a right circular cylinder of diameter 36m and height 44dm surmounted by a right circular cone. Find the cost of canvas at Rs.3.50 per m2 ?

Ex. 14.20 | Q 4 | Page 60

A toy is in the form of a cone surmounted on a hemisphere. The diameter of the base and the height of cone are 6cm and 4cm. determine surface area of toy?

Ex. 14.20 | Q 5 | Page 60

A solid is in the form of a right circular cylinder, with a hemisphere at one end and a cone at the other end. The radius of the common base is 3.5 cm and the heights of the cylindrical and conical portions are 10 cm. and 6 cm, respectively. Find the total surface area of the solid. (Use n =`22/7`)

Ex. 14.20 | Q 6 | Page 60

A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The radius and height of the cylindrical part are 5 cm and 13 cm respectively.The radii of the hemispherical and conical parts are the same as that of the cylindrical part.Find the surface area of the toy if the total height of the toy is 30 cm.

Ex. 14.20 | Q 7 | Page 60

A cylindrical tub of radius 5 cm and length 9.8 cm is full of water. A solid in the form of a right circular cone mounted on a hemisphere is immersed in the tub. If the radius of the hemisphere is immersed in the tub. If the radius of the hemi-sphere is 3.5 cm and height of the cone outside the hemisphere is 5 cm, find the volume of the water left in the tub (Take π = 22/7)

Ex. 14.20 | Q 8 | Page 61

A circus tent has cylindrical shape surmounted by a conical roof. The radius of the cylindrical base is 20 m. The heights of the cylindrical and conical portions are 4.2 m and 2.1 m respectively. Find the volume of the tent.

Ex. 14.20 | Q 9 | Page 61

A petrol tank is a cylinder of base diameter 21 cm and length 18 cm fitted with conical ends each of axis length 9 cm. Determine the capacity of the tank.

Ex. 14.20 | Q 10 | Page 61

A conical hole is drilled in a circular cylinder of height 12 cm and base radius 5 cm. The height and the base radius of the cone are also the same. Find the whole surface and volume of the remaining cylinder.

Ex. 14.20 | Q 11 | Page 61

A tent is in the form of a cylinder of diameter 20 m and height 2.5 m, surmounted by a cone of equal base and height 7.5 m. Find the capacity of the tent and the cost of the canvas at Rs 100 per square metre.

Ex. 14.20 | Q 12 | Page 61

A boiler is in the form of a cylinder 2 m long with hemispherical ends each of 2 metre diameter. Find the volume of the boiler.

Ex. 14.20 | Q 13 | Page 61

A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylinder is `14/3`  m and the diameter of hemisphere is 3.5 m. Calculate the volume and the internal surface area of the solid.

Ex. 14.20 | Q 14 | Page 61

A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and the radius of each of the hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs 10 per dm2 .

Ex. 14.20 | Q 15 | Page 61

A cylindrical vessel of diameter 14cm and height 42cm is fixed symmetrically inside a similar vessel of diameter 16cm and height 42 . cm The total space between two vessels is filled with cork dust for heat insulation purpose. How many cubic cms of cork dust will be
required?

Ex. 14.20 | Q 16 | Page 61

A cylindrical road roller made of iron is 1 m long, Its internal diameter is 54 cm and the thickness of the iron sheet used in making the roller is 9 cm. Find the mass of the roller, if 1 cm3 of iron has 7.8 gm mass. (Use π = 3.14)

Ex. 14.20 | Q 17 | Page 61

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. [Use π = 22/7]

Ex. 14.20 | Q 18 | Page 61

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy [Use π =22/7]

Ex. 14.20 | Q 19 | Page 61

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 m2. If the pipe is made of 99 cm3 of metal, find the outer and inner radii of the pipe.

Ex. 14.20 | Q 20 | Page 61

A right circular cylinder having diameter 12 cm and height 15 cm is full ice-cream. The ice-cream is to be filled in cones of height 12 cm and diameter 6 cm having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.

Ex. 14.20 | Q 21 | Page 61

A solid iron pole having cylindrical portion 110 cm high and of base diameter 12 cm is surmounted by a cone 9 cm high. Find the mass of the pole, given that the mass of 1 cm3of iron is 8 gm.

Ex. 14.20 | Q 22 | Page 61

A solid toy is in the form of a hemisphere surmounted by a right circular cone. height of the cone is 2 cm and the diameter of the base is 4 cm. If a right circular cylinder circumscribes the toy, find  how much more space it will cover.

Ex. 14.20 | Q 23 | Page 61

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm. Use [Π = 22/7]

Ex. 14.20 | Q 24 | Page 62

A cylindrical vessel with internal diameter 10 cm and height 10.5 cm is full of water. A solid cone of base diameter 7 cm and height 6 cm is completely immersed in water. Find the value of water (i) displaced out of the cylinder (ii) left in the cylinder. (Take π 22/7)

Ex. 14.20 | Q 25 | Page 62

A hemispherical depression is cut out from one face of a cubical wooden block of edge 21 cm, such that the diameter of the hemisphere is equal to the edge of the cube. Determine the volume and total surface area of the remaining block.

Ex. 14.20 | Q 26 | Page 62

A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of the base of the cone is 21 cm and its volume is `2/3` of the volume of hemisphere, calculate the height of the cone and the surface area of the toy.
 `(use pi = 22/7)`

Ex. 14.20 | Q 27 | Page 62

A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them is being 3.5 cm and the total height of solid is 9.5 cm. Find the volume of the solid. (Use π = 22/7).

Ex. 14.20 | Q 28 | Page 62

A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy. `[\text{Use}pi 22/7]`

Ex. 14.20 | Q 29 | Page 62

The largest possible sphere is carved out of a wooden solid cube of side 7 em. Find the volume of the wood left. (Use\[\pi = \frac{22}{7}\]).

Ex. 14.20 | Q 30 | Page 62

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid [take π=22/7]

Ex. 14.20 | Q 31 | Page 62

The largest cone is curved out from one face of solid cube of side 21 cm. Find the volume of the remaining solid. 

Ex. 14.20 | Q 32 | Page 62

A solid wooden toy is in the form of a hemisphere surrounded by a cone of same radius. The radius of hemisphere is 3.5 cm and the total wood used in the making of toy is 166 `5/6`  cm3. Find the height of the toy. Also, find the cost of painting the hemispherical part of the toy at the rate of Rs 10 per cm2 .[Use`pi=22/7`]

Ex. 14.20 | Q 33 | Page 62

In Fig. 5, from a cuboidal solid metallic block, of dimensions 15cm ✕ 10cm ✕ 5cm, a cylindrical hole of diameter 7 cm is drilled out. Find the surface area of the remaining block [Use

`pi=22/7`]

Ex. 14.20 | Q 34 | Page 62

 A building is in the form of a cylinder surmounted by a hemi-spherical vaulted dome and contains  \[41\frac{19}{21} m^3\] of air. If the internal diameter of dome is equal to its total height  above the floor , find the height of the building ?

Ex. 14.20 | Q 35 | Page 62

A pen stand made of wood is in the shape of a cuboid with four conical depression and a cubical depression to hold the pens and pins , respectively . The dimension of the cuboid are  \[10 cm \times 5 cm \times 4 cm\].

The radius of each of the conical depression is 0.5 cm and the depth is 2.1 cm . The edge of the cubical depression is 3 cm . Find the volume of the wood in the entire stand. 

Ex. 14.20 | Q 36 | Page 63

A building is in the form of a cylinder surrounded by a hemispherical dome. The base diameter of the dome is equal to  \[\frac{2}{3}\] of the total height of the building . Find the height of the building , if it contains  \[67\frac{1}{21} m^3\].

Ex. 14.20 | Q 37 | Page 63

A solid toy s in the form of a hemisphere  surrounded by a right circular cone . The height of cone is 4 cm and the diameter of the base is 8 cm . Determine the volume of the toy. If a cube circumscribes the toy , then find the difference of the volumes of cube and the toy .

Ex. 14.20 | Q 38 | Page 63

A circus tent is in the shape of cylinder surmounted by a conical top of same diameter. If their common diameter is 56 m, the height of the cylindrical part is 6 m and the total height of the tent above the ground is 27 m, find the area of the canvas used in making the tent.

Chapter 14: Surface Areas and Volumes Exercise 14.30 solutions [Pages 78 - 80]

Ex. 14.30 | Q 1 | Page 78

A bucket has top and bottom diameter of 40 cm and 20 cm respectively. Find the volume of the bucket if its depth is 12 cm. Also, find the cost of tin sheet used for making the bucket at the rate of Rs. 1.20 per dm. (Use π = 3.14)

Ex. 14.30 | Q 2 | Page 78

A frustum of a right circular cone has a diameter of base 20 cm, of top 12 cm, and height 3 cm. Find the area of its whole surface and volume.

Ex. 14.30 | Q 3 | Page 78

The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. find the curved surface area of the frustum.

Ex. 14.30 | Q 4 | Page 78

The perimeters of the ends of a frustum of a right circular cone are 44 cm and 33 cm. If the height of the frustum be 16 cm, find its volume, the slant surface and the total surface.

 

Ex. 14.30 | Q 5 | Page 78

If the radii of the circular ends of a conical bucket which is 45 cm high be 28 cm and 7 cm, find the capacity of the bucket. (Use π = 22/7).

Ex. 14.30 | Q 6 | Page 78

The height of a cone is 20 cm. A small cone is cut off from the top by a plane parallel to the base. If its volume be 1/125 of the volume of the original cone, determine at what height above the base the section is made.

Ex. 14.30 | Q 7 | Page 78

If the radii of circular ends of a bucket 24cm high are 5cm and 15cm. find surface area of
bucket?

Ex. 14.30 | Q 8 | Page 79

The radii of the circular bases of a frustum of a right circular cone are 12 cm and 3 cm and the height is 12 cm. Find the total surface area and the volume of the frustum.

Ex. 14.30 | Q 9 | Page 79

A tent consists of a frustum of a cone capped by a cone. If the radii of the ends of the frustum be 13 m and 7 m , the height of the frustum be 8 m and the slant height of the conical cap be 12 m, find the canvas required for the tent. (Take : π = 22/7)

Ex. 14.30 | Q 10 | Page 79

A milk container of height 16 cm is made of metal sheet in the form of a frustum of a cone with radii of its lower and upper ends as 8 cm and 20 cm respectively . Find the cost of milk at the rate of ₹44 per litre which the container can hold. 

Ex. 14.30 | Q 11 | Page 79

A bucket is in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm respectively. Find the capacity and surface area of the  bucket. Also, find the cost of milk which can completely fill the container , at thr rate of ₹25 per litre. (Use \[\pi = 3 . 14) .\]

Ex. 14.30 | Q 12 | Page 79

A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm3 of water.The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of the metal sheet used in its making. (Use 𝜋 = 3.14).

Ex. 14.30 | Q 13 | Page 79

A bucket made of aluminum sheet is of height 20cm and its upper and lower ends are of radius 25cm an 10cm, find cost of making bucket if the aluminum sheet costs Rs 70 per
100 cm2

Ex. 14.30 | Q 14 | Page 79

The radii of the circular ends of a solid frustum of a cone are 33 cm and 27 cm and its slant height is 10 cm. Find its total surface area.

Ex. 14.30 | Q 15 | Page 79

A bucket made up of a metal sheet is in form of a frustum of cone of height 16cm with diameters of its lower and upper ends as 16cm and 40cm. find the volume of bucket. Also find cost of bucket if the cost of metal sheet used is Rs 20 per 100 cm2

Ex. 14.30 | Q 16 | Page 79

A solid is in the shape of a frustum of a cone. The diameter of two circular ends are 60cm and 36cm and height is 9cm. find area of its whole surface and volume?

Ex. 14.30 | Q 17 | Page 79

A milk container is made of metal sheet in the shape of frustum of a cone whose volume is 10459 `3/7` cm3. The radii of its lower and upper circular ends are 8cm and 20cm. find the cost of metal sheet used in making container at rate of  Rs 1.4  per cm2?

Ex. 14.30 | Q 18 | Page 79

A solid cone of base radius 10 cm is cut into two part through the mid-point of its height, by a plane parallel to its base. Find the ratio in the volumes of two parts of the cone.

Ex. 14.30 | Q 19 | Page 79

A bucket open at the top, and made up of a metal sheet is in the form of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs 10 per 100 cm2. [Use π = 3.14] 

Ex. 14.30 | Q 20 | Page 79

In Fig. 4, from the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid. (Use `pi=22/7` and `sqrt5=2.236`)

Ex. 14.30 | Q 21 | Page 80

The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.

Ex. 14.30 | Q 22 | Page 80

A bucket, made of metal sheet, is in the form of a cone whose height is 35 cm and radii of circular ends are 30 cm and 12 cm. How many litres of milk it contains if it is full to the brim? If the milk is sold at Rs 40 per litre, find the amount received by the person.

Ex. 14.30 | Q 23 | Page 80

A reservoir in the form of the frustum of a right circular cone contains 44 × 107 litres of water which fills it completely. The radii of the bottom and top of the reservoir are 50 metres and 100 metres respectively. Find the depth of water and the lateral surface area of the reservoir. (Take: π = 22/7)

Chapter 14: Surface Areas and Volumes Exercise 14.30 solutions [Pages 80 - 85]

Ex. 14.30 | Q 1 | Page 80

A metallic sphere 1 dm in diameter is beaten into a circular sheet of uniform thickness equal to 1 mm. Find the radius of the sheet.

Ex. 14.30 | Q 2 | Page 80

Three solid spheres of radii 3, 4 and 5 cm respectively are melted and converted into a single solid sphere. Find the radius of this sphere.

Q 3 | Page 81

A spherical shell of lead, whose external diameter is 18 cm, is melted and recast into a right circular cylinder, whose height is 8 cm and diameter 12 cm. Determine the internal diameter of the shell.

Q 4 | Page 81

A well with 10 m inside diameter is dug 8.4 m deep. Earth taken out of it is spread all around it to a width of 7.5 m to form an embankment. Find the height of the embankment.

Q 5 | Page 81

In the middle of a rectangular field measuring 30 m × 20 m, a well of 7 m diameter and 10 m depth is dug. The earth so removed is evenly spread over the remaining part of the field. Find the height through which the level of the field is raised.

Q 6 | Page 81

The inner and outer radii of a hollow cylinder are 15 cm and 20 cm, respectively. The cylinder is melted and recast into a solid cylinder of the same height. Find the radius of the base of new cylinder.

Q 7 | Page 81

Two cylindrical vessels are filled with oil. Their radii are 15 cm, 12 cm and heights 20 cm, 16 cm respectively. Find the radius of a cylindrical vessel 21 cm in height, which will just contain the oil of the two given vessels.

Q 8 | Page 81

A cylindrical bucket 28 cm in diameter and 72 cm high is full of water. The water is emptied into a rectangular tank 66 cm long and 28 cm wide. Find the height of the water level in the tank.

Q 9 | Page 81

A cubic cm of gold is drawn into a wire 0.1 mm in diameter, find the length of the wire.

Q 10 | Page 81

A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

Q 11 | Page 81

A conical vessel whose internal radius is 10 cm and height 48 cm is full of water. Find the volume of water. If this water is poured into a cylindrical vessel with internal radius 20 cm, find the height to which the water level rises in it.

Q 12 | Page 81

The vertical height of a conical tent is 42 dm and the diameter of its base is 5.4 m. Find the number of persons it can accommodate if each person is to be allowed 29.16 cubic dm.

Q 13 | Page 81

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.

Q 14 | Page 81

A sphere of diameter 5 cm is dropped into a cylindrical vessel partly filled with water. The diameter of the base of the vessel is 10 cm. If the sphere is completely submerged, by how much will the level of water rise?

Q 15 | Page 81

A spherical ball of iron has been melted and made into smaller balls. If the radius of each smaller ball is one-fourth of the radius of the original one, how many such balls can be made?

Q 16 | Page 81

Find the depth of a cylindrical tank of radius 28 m, if its capacity is equal to that of a rectangular tank of size 28 m × 16 m × 11 m.

Q 17 | Page 81

A hemispherical bowl of internal radius 15 cm contains a liquid. The liquid is to be filled into cylindrical-shaped bottles of diameter 5 cm and height 6 cm. How many bottles are necessary to empty the bowl?

Q 18 | Page 81

In a cylindrical vessel of diameter 24 cm, filled up with sufficient quantity of water, a solid spherical ball of radius 6 cm is completely immersed. Find the increase in height of water level.

Q 19 | Page 81

A hemisphere of lead of radius 7 cm is cast into a right circular cone of height 49 cm. Find the radius of the base.

Q 20 | Page 82

A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter  4 \[\frac{2}{3}\] cm and height 3 cm. Find the number of cones so formed.

Q 21 | Page 82

The diameter of a copper sphere is 18 cm. The sphere is melted and is drawn into a long wire of uniform circular cross-section. If the length of the wire is 108 m, find its diameter.

Q 22 | Page 82

A hemisphere of lead of radius 7 cm is cast into a right circular cone of height 49 cm. Find the radius of the base.

Q 23 | Page 82

A metallic sphere of radius 10.5 cm is melted and thus recast into small cones, each of radius 3.5 cm and height 3 cm. Find how many cones are obtained.

Q 24 | Page 82

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1 : 2 : 3.

Q 25 | Page 82

A hollow sphere of internal and external diameters 4 and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.

Q 26 | Page 82

The largest sphere is carved out of a cube of side 10.5 cm. Find the volume of the sphere.

Q 27 | Page 82

Find the weight of a hollow sphere of metal having internal and external diameters as 20 cm and 22 cm, respectively if 1m3 of metal weighs 21g.

Q 28 | Page 82

A solid sphere of radius 'r' is melted and recast into a hollow cylinder of uniform thickness. If the external radius  of the base of the cylinder is 4 cm, its height 24 cm and thickness 2 cm, find the value of 'r'.

Q 29 | Page 82

Lead spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water and are fully submerged. If the diameter of the beaker is 18 cm and water rises by 40 cm. find the number of lead spheres dropped in the water.

Q 30 | Page 82

The height of a solid cylinder is 15 cm and the diameter of its base is 7 cm. Two equal conical holes each of radius 3 cm and height 4 cm are cut off. Find the volume of the remaining solid.

Q 31 | Page 82

A solid is composed of a cylinder with hemispherical ends. If the length of the whole solid is 108 cm and the diameter of the cylinder is 36 cm, find the cost of polishing the surface at the rate of 7 paise per cm2 .

Q 32 | Page 82

The surface area of a sphere is the same as the curved surface area of a cone having the radius of the base as 120 cm and height 160 cm. Find the radius of the sphere.

Q 33 | Page 82

A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ratio 8 : 5, determine the ratio of the radius of the base to the height of either of them.

Q 34 | Page 82

A rectangular vessel of dimensions 20 cm × 16 cm × 11 cm is full of water. This water is poured into a conical vessel. The top of the conical vessel has its radius 10 cm. If the conical vessel is filled completely, determine its height.

Q 35 | Page 82

If r1 and r2 be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new sphere is \[\left( r_1^3 + r_2^3 \right)^\frac{1}{3}\].

Q 36 | Page 82

A solid metal sphere of 6 cm diameter is melted and a circular sheet of thickness 1 cm is prepared. Determine the diameter of the sheet.

Q 37 | Page 82

A hemispherical tank full of water is emptied by a pipe at the rate of \[\frac{25}{7}\] litres per second. How much time will it take to half-empty the tank, If the tank is 3 metres in diameter?

 
Q 38 | Page 83

Find the number of coins, 1.5 cm is diameter and 0.2 cm thick, to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

Q 39 | Page 83

The radius of the base of a right circular cone of semi-vertical angle α is r. Show that its volume is \[\frac{1}{3} \pi r^3\] cot α and curved surface area is πr2 cosec α.

 
Q 40 | Page 83

An iron pillar consists of a cylindrical portion 2.8 m high and 20 cm in diameter and a cone 42 cm high is surmounting it. Find the weight of the pillar, given that 1 cubic cm of iron weighs 7.5 gm.

Q 41 | Page 83

A circus tent is cylindrical to a height of 3 metres and conical above it. If its diameter is 105 m and the slant height of the conical portion is 53 m, calculate the length of the canvas 5 m wide to make the required tent.

Q 42 | Page 83

Height of a solid cylinder is 10 cm and diameter 8 cm. Two equal conical hole have been made from its both ends. If the diameter of the holes is 6 cm and height 4 cm, find (i) volume of the cylinder, (ii) volume of one conical hole, (iii) volume of the remaining solid.

 
Q 43 | Page 83

The height of a solid cylinder is 15 cm and the diameter of its base is 7 cm. Two equal conical holes each of radius 3 cm and height 4 cm are cut off. Find the volume of the remaining solid.

Q 44 | Page 83

A solid is composed of a cylinder with hemispherical ends. If the length of the whole solid is 108 cm and the diameter of the cylinder is 36 cm, find the cost of polishing the surface at the rate of 7 paise per cm2 .

Q 45 | Page 83

The largest sphere is to be curved out of a right circular of radius 7cm and height 14cm. find volume of sphere?

Q 46 | Page 83

A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of cylinder is 24 m. The height of the cylindrical portion is 11 m while the vertex of the cone is 16 m above the ground. Find the area of canvas required for the tent.

Q 47 | Page 83

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy [Use π =22/7]

Q 48 | Page 83

A cylindrical container is filled with ice-cream, whose diameter is 12 cm and height is 15 cm. the whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is twice the diameter of its base, find the diameter of the ice-cream.

Q 49 | Page 83

Find the volume of a solid in the form of a right circular cylinder with hemi-spherical ends whose total length is 2.7 m and the diameter of each hemi-spherical end is 0.7 m.

Q 50 | Page 83

A tent of height 8.25 m is in the form of a right circular cylinder with diameter of base 30 m and height 5.5 m, surmounted by a right circular cone of the same base. Find the cost of the canvas of the tent at the rate of Rs 45 per m2.

Q 51 | Page 83

An iron pole consisting  of a cylindrical portion 110 cm high and of base diameter 12 cm is surmounted by a cone 9 cm high. Find the mass of the pole, given that 1 cm3 of iron has 8 gram mass approximately. (Use : π = 355/115)

Q 52 | Page 83

The interior of a building is in the form of a cylinder of base radius 12 m and height 3.5 m, surmounted by a cone of equal base and slant height 12.5 m. Find the internal curved surface area and the capacity of the building.

Q 53 | Page 84

A right angled triangle with sides 3 cm and 4 cm is revolved around its hypotenuse. Find the volume of the double cone thus generated.

Q 54 | Page 84

A toy is in the form of a cone surmounted on a hemisphere. The diameter of the base and the height of cone are 6cm and 4cm. determine surface area of toy?

Q 55 | Page 84

Find the mass of a 3.5 m long lead pipe, if the external diameter of the pipe is 2.4 cm, thickness of the metal is 2 mm and the mass of 1 cm3 of lead is 11.4 grams.

Q 56 | Page 84

A solid is in the form of a cylinder with hemispherical ends. Total height of the solid is 19 cm and the diameter of the cylinder is 7 cm. Find the volume and total surface area of the solid.

Q 57 | Page 84

A golf ball has diameter equal to 4.2 cm. Its surface has 200 dimples each of radius 2 mm. Calculate the total surface area which is exposed to the surroundings assuming that the dimples are hemispherical.

Q 58 | Page 84

The radii of the ends of a bucket of height 24 cm are 15 cm and 5 cm. Find its capacity. (Take π = 22/7)

Q 59 | Page 84

The radii of the ends of a bucket 30 cm high are 21 cm and 7 cm. Find its capacity in litres and the amount of sheet required to make this bucket.

Q 60 | Page 84

The radii of the ends of a frustum of a right circular cone are 5 metres and 8 metres and its lateral height is 5 metres. Find the lateral surface and volume of the frustum.

Q 61 | Page 84

A frustum of a cone is 9 cm thick and the diameters of its circular ends are 28 cm and 4 cm. Find the volume and lateral surface area of the frustum.
(Take π = 22/7).

Q 62 | Page 84

A bucket is in the form of a frustum of a cone and holds 15.25 litres of water. The diameters of the top and bottom are 25 cm and 20 cm respectively. Find its height and area of tin used in its construction.

Q 63 | Page 84

If a cone of radius 10 cm is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.

Q 64 | Page 84

A tent is of the shape of a right circular cylinder upto a height of 3 metres and then becomes a right circular cone with a maximum height of 13.5 metres above the ground. Calculate the cost of painting the inner side of the tent at the rate of Rs 2 per square metre, if the radius of the base is 14 metres.

Q 65 | Page 84

An oil funnel of tin sheet consists of a cylindrical portion 10 cm long attached to a frustum of a cone. If the total height be 22 cm, the diameter of the cylindrical portion 8 cm and the diameter of the top of the funnel 18 cm, find the area of the tin required.(Use π = 22/7).

Q 66 | Page 84

A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into toys with the shape of a right circular cone mounted on a hemisphere of radius 3 cm.If the height of the toy is 12 cm, find the number of toys so formed.

Q 67 | Page 84

A container open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends, as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container. at the rate of 21 per litre. [use π=22/7]

Q 68 | Page 85

A cone of maximum size is carved out from a cube of edge 14 cm . Find the surface area of the cone and of the remaining solid left out after the cone carved out .

Q 69 | Page 85

A cone of radius 4 cm is divided into two parts by drawing a plane through the mid point of its axis and parallel to its base . Compare the volumes of two parts. 

Q 70 | Page 85

A wall 24 m , 0.4 m thick and 6 m high is constructed with the bricks each of dimensions 25 cm  \[\times\] 16 cm \[\times\] 10 cm . If the mortar occupies  \[\frac{1}{10}th\] of the volume of the wall, then find the number of bricks used in constructing the wall.

 
Q 71 | Page 85

A bucket is in the form of  a frustum of a cone and holds 28.490 litres of water . The radii of the top and bottom are 28 cm and 21 cm respectively . Find the height of the bucket . 

Q 72 | Page 85

Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm containing some water . Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6 cm .

Q 73 | Page 85

Two cones with same base radius 8 cm and height 15 cm are joined together along their bases. Find the surface area of the shape formed.

Q 74 | Page 85

From a solid cube of side 7 cm , a conical cavity of height 7 cm and radius 3 cm is hollowed out . Find the volume of the remaining solid.

Q 75 | Page 85

Two solid cones and B are placed in a cylindrical tube as shown in fig .16.76. The ratio of their capacities are 2: 1 . Find the heights and capacities of the cones . Also, find the volume of the remaining portion of the cylinder.

Q 76 | Page 85

An icecream cone full of icecream having radius 5 cm and height 10 cm as shown in fig. 16.77. Calculate the volume of icecream , provided that its 1/ 6 part is left unfilled with icecream .

Chapter 14: Surface Areas and Volumes solutions [Pages 86 - 87]

Q 1 | Page 86

The radii of the base of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3. What is the ratio of their volumes?

Q 2 | Page 86

If the heights of two right circular cones are in the ratio 1 : 2 and the perimeters of their bases are in the ratio 3 : 4, what is the ratio of their volumes?

Q 3 | Page 86

If a cone and a sphere have equal radii and equal volumes. What is the ratio of the diameter of the sphere to the height of the cone?

Q 4 | Page 86

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes?

Q 5 | Page 86

The radii of two cylinders are in the ratio 3 : 5 and their heights are in the ratio 2 : 3. What is the ratio of their curved surface areas?

Q 6 | Page 86

Two cubes have their volumes in the ratio 1 : 27. What is the ratio of their surface areas?

Q 7 | Page 86

Two right circular cylinders of equal volumes have their heights in the ratio 1 : 2. What is the ratio of their radii ?

Q 8 | Page 87

If the volumes of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then write the ratio of their weights.

Q 9 | Page 87

A sphere and a cube have equal surface areas. What is the ratio of the volume of the sphere to that of the cube?

Q 10 | Page 87

What is the ratio of the volume of a cube to that of a sphere which will fit inside it?

Q 11 | Page 87

What is the ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height?

Q 12 | Page 87

A sphere of maximum volume is cut-out from a solid hemisphere of radius r, what is the ratio of the volume of the hemisphere to that of the cut-out sphere?

Q 13 | Page 87

A metallic hemisphere is melted and recast in the shape of a cone with the same base radius R as that of the hemisphere. If H is the height of the cone, then write the values of \[\frac{H}{R} .\]

Q 14 | Page 87

A right circular cone and a right circular cylinder have equal base and equal height. If the radius of the base and height are in the ratio 5 : 12, write the ratio of the total surface area of the cylinder to that of the cone.

Q 15 | Page 87

A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ratio of their volumes?

Q 16 | Page 87

The radii of two cones are in the ratio 2 : 1 and their volumes are equal. What is the ratio of their heights?

Q 17 | Page 87

Two cones have their heights in the ratio 1 : 3 and radii 3 : 1. What is the ratio of their volumes?

Q 18 | Page 87

A hemisphere and a cone have equal bases. If their heights are also equal, then what is the ratio of their curved surfaces?

Q 19 | Page 87

If r1 and r2 denote the radii of the circular bases of the frustum of a cone such that r1 > r2, then write the ratio of the height of the cone of which the frustum is a part to the height fo the frustum.

Q 20 | Page 87

If the slant height of the frustum of a cone is 6 cm and the perimeters of its circular bases are 24 cm and 12 cm respectively. What is the curved surface area of the frustum?

Q 21 | Page 87

If the areas of circular bases of a frustum of a cone are 4 cm2 and 9 cm2 respectively and the height of the frustum is 12 cm. What is the volume of the frustum?

Q 22 | Page 87

The surface area of a sphere is 616 cm2 . Find its radius.

Q 23 | Page 87

A cylinder and a cone are of the same base radius and of same height. Find the ratio of the value of the cylinder to that of the cone.

Q 24 | Page 87

The slant height of the frustum of a cone is 5 cm. If the difference between the radii of its two circular ends is 4 cm, write the height of the frustum.

Q 25 | Page 87

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Chapter 14: Surface Areas and Volumes solutions [Pages 88 - 91]

Q 1 | Page 88

The diameter of a sphere is 6 cm. It is melted and drawn in to a wire of diameter 2 mm. The length of the wire is

  • 12 m

  • 18 m

  •  36 m

  • 66 m

Q 2 | Page 88

A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. The number of such cones is

  • 63

  • 126

  • 21

  • 130

Q 3 | Page 88

A solid is hemispherical at the bottom and conical above. If the surface areas of the two parts are equal, then the ratio of its radius and the height of its conical part is

  •  1 : 3

  • 1 : \[\sqrt{3}\]

  • 1 : 1

  • \[\sqrt{3}\] :1

Q 4 | Page 88

A solid sphere of radius r is melted and cast into the shape of a solid cone of height r, the radius of the base of the cone is

  • 2r

  •  3r

  • r

  • 4r

Q 5 | Page 88

The material of a cone is converted into the shape of a cylinder of equal radius. If height of the cylinder is 5 cm, then height of the cone is

  • 10 cm

  • 15 cm

  • 18 cm

  • 24 cm

Q 6 | Page 88

A circus tent is cylindrical to a height of 4 m and conical above it. If its diameter is 105 m and its slant height is 40 m, the total area of the canvas required in m2 is

  • 1760

  • 2640

  • 3960

  • 7920

Q 7 | Page 88

The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:

  • 3

  • 4

  • 5

  • 6

Q 8 | Page 88

A sphere of radius 6 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 8 cm. If the sphere is submerged completely, then the surface of the water rises by

  • 4.5 cm

  • 3

  • 4 cm

  • 2 cm

Q 9 | Page 88

If the radii of the circular ends of a bucket of height 40 cm are of lengths 35 cm and 14 cm, then the volume of the bucket in cubic centimeters, is

  • 60060

  •  80080

  • 70040

  • 80160

Q 10 | Page 88

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, the ratio of the volumes of the upper part and the cone is

  • 1 : 2

  • 1: 4

  •  1 : 6

  • 1 : 8

Q 11 | Page 88

The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be \[\frac{1}{27}\] of the volume of the given cone, then the height above the base at which the section has been made, is

  • 10 cm

  • 15 cm

  • 20 cm

  • 25 cm

Q 12 | Page 88

A solid consists of a circular cylinder with an exact fitting right circular cone placed at the top. The height of the cone is h. If the total volume of the solid is 3 times the volume of the cone, then the height of the circular is

  •  2h

  • \[\frac{2h}{3}\]

  • \[\frac{3h}{2}\]

  •  4h

Q 13 | Page 89

A reservoir is in the shape of a frustum of a right circular cone. It is 8 m across at the top and 4 m across at the bottom. If it is 6 m deep, then its capacity is

  • 176 m3

  • 196 m3

  • 200 m3

  • 110 m3

Q 14 | Page 89

Water flows at the rate of 10 metre per minute from a cylindrical pipe 5 mm in diameter. How long will it take to fill up a conical vessel whose diameter at the base is 40 cm and depth 24 cm?

  • 48 minutes 15 sec

  • 51 minutes 12 sec

  • 52 minutes 1 sec

  •  55 minutes

Q 15 | Page 89

A cylindrical vessel 32 cm high and 18 cm as the 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, the radius of its base is

  • 12 cm

  •  24 cm

  • 36 cm

  •  48 cm

Q 16 | Page 89

The curved surface area of a right circular cone of height 15 cm and base diameter 16 cm is

  • 60π cm2

  • 68π cm2

  • 120π cm2

  • `136 pi cm^3`

Q 17 | Page 89

A right triangle with sides 3 cm, 4 cm and 5 cm is rotated about the side of 3 cm to form a cone. The volume of the cone so formed is

  • 12π cm3

  • 15π cm3

  • 16π cm3

  •  20π cm3

Q 18 | Page 89

The curved surface area of a cylinder is 264 m2 and its volume is 924 m3. The ratio of its diameter to its height is

  • 3 : 7

  •  7 : 3

  • 6 : 7

  • 7 : 6

Q 19 | Page 89

A cylinder with base radius of 8 cm and height of 2 cm is melted to form a cone of height 6 cm. The radius of the cone is

  • 4 cm

  • 5 cm

  • 6 cm

  •  8 cm

Q 20 | Page 89

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

  • 1 : 2

  • 2 : 3

  • 9 : 16

  •  16 : 9

Q 21 | Page 89

If three metallic spheres of radii 6 cm, 8 cm and 10 cm are melted to form a single sphere, the diameter of the sphere is

  • 12 cm

  •  24 cm

  •  30 cm

  • 36 cm

Q 22 | Page 89

The surface area of a sphere is same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is

  • 3 cm

  • 4 cm

  • 6 cm

  • 12 cm

Q 23 | Page 89

The volume of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is

  • \[\frac{4}{3}\pi\]

  • \[\frac{10}{3}\pi\]

  • 5\[\pi\]

  • \[\frac{20}{3}\pi\]

Q 24 | Page 89

A cylindrical vessel of radius 4 cm contains water. A solid sphere of radius 3 cm is lowered into the water until it is completely immersed. The water level in the vessel will rise by

  • \[\frac{2}{9}\] cm

  • \[\frac{4}{9}\]cm

  • \[\frac{9}{4}\] cm

  • \[\frac{9}{2}\]cm

Q 25 | Page 90

12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is

  • \[\sqrt{3}\] cm

  • 2cm

  • 3cm

  • 4cm

Q 26 | Page 90

A solid metallic spherical ball of diameter 6 cm is melted and recast into a cone with diameter of the base as 12 cm. The height of the cone is

  • 2 cm

  • 3 cm

  • 6 cm

Q 27 | Page 90

A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. The height of the cone is

  • 12 cm

  • 14 cm

  • 15 cm

  • 18 cm

Q 28 | Page 90

A solid piece of iron of dimensions 49 × 33 × 24 cm is moulded into a sphere. The radius of the sphere is

  • 21 cm

  • 28 cm

  • 35 cm

  • none of these

Q 29 | Page 90

The ratio of lateral surface area to the total surface area of a cylinder with base diameter 1.6 m and height 20 cm is

  • 1 : 7

  • 1 : 5

  • 7 : 1

  • 8 : 1

Q 30 | Page 90

A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is

  • 2h

  • \[\frac{3h}{2}\]

  • \[\frac{h}{2}\]

  • \[\frac{2h}{3}\]

Q 31 | Page 90

The maximum volume of a cone that can be carved out of a solid hemisphere of radius r is

  • \[3 \pi r^2\]

  • `1/3pir^3`

     

  • \[\frac{\pi r^2}{3}\]

  • \[3 \pi r^3\]

Q 32 | Page 90

The radii of two cylinders are in the ratio 3 : 5. If their heights are in the ratio 2 : 3, then the ratio of their curved surface areas is

  • 2 : 5

  • 5 : 2

  • 2 : 3

  • 3 : 5

Q 33 | Page 90

A right circular cylinder of radius r and height h (h = 2r) just encloses a sphere of diameter

  • h

  • r

  • 2r

  • 2h

Q 34 | Page 90

The radii of the circular ends of a frustum are 6 cm and 14 cm. If its slant height is 10 cm, then its vertical height is

  • 6 cm

  • 8 cm

  • 4 cm

  • 7 cm

Q 35 | Page 90

The height and radius of the cone of which the frustum is a part are h1 and r1 respectively. If h2 and r2 are the heights and radius of the smaller base of the frustum respectively and h2 : h1 = 1 : 2, then r2 : r1 is equal to

  • 1 : 3

  • 1 : 2

  • 2 : 1

  •  3 : 1

Q 36 | Page 90

The diameters of the ends of a frustum of a cone are 32 cm and 20 cm. If its slant height is 10 cm, then its lateral surface area is

  • 321 π cm2

  • 300 π cm2

  • 260 π cm2

  • 250 π cm2

Q 37 | Page 90

A solid frustum is of height 8 cm. If the radii of its lower and upper ends are 3 cm and 9 cm respectively, then its slant height is

  • 15 cm

  • 12 cm

  •  10 cm

  • 17 cm

Q 38 | Page 91

The radii of the ends of a bucket 16 cm height are 20 cm and 8 cm. The curved surface area of the bucket is

  • 1760 cm2

  •  2240 cm2

  •  880 cm2

  •  3120 cm2

Q 39 | Page 91

The diameters of the top and the bottom portions of a bucket are 42 cm and 28 cm respectively. If the height of the bucket is 24 cm, then the cost of painting its outer surface at the rate of 50 paise / cm2 is

  • Rs. 1582.50

  • Rs. 1724.50

  •  Rs. 1683

  •  Rs. 1642

Q 40 | Page 91

If four times the sum of the areas of two circular faces of a cylinder of height 8 cm is equal to twice the curve surface area, then diameter of the cylinder is

  • 4 cm

  • 8 cm

  • 2 cm

  • 6 cm

Q 41 | Page 91

If the radius of the base of a right circular cylinder is halved, keeping the height the same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is:

  •  1 : 2

  •  2 : 1

  •  1 : 4

  •  4 : 1

Q 42 | Page 91

A metalic solid cone is melted to form a solid cylinder of equal radius. If the height of the cylinder is 6 cm, then the height of the cone was

  • 10 cm

  • 12 cm

  • 18 cm

  • 24 cm

Q 43 | Page 91

A rectangular sheet of paper 40 cm × 22 cm, is rolled to form a hollow cylinder of height 40 cm. The radius of the cylinder (in cm) is

  •  3.5

  • 7

  • \[\frac{80}{7}\]

  • 5

Q 44 | Page 91

The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:

  • 3

  • 4

  • 5

  • 6

Q 45 | Page 91

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

  • 1 : 2

  • 2 : 3

  • 9 : 16

  •  16 : 9

Q 46 | Page 91

A right circular cylinder of radius r and height h (h = 2r) just encloses a sphere of diameter

  • h

  • r

  • 2r

  • 2h

Q 47 | Page 91

In a right circular cone , the cross-section made by a plane parallel to the base is a 

  •  circle

  •  frustyum of a cone

  • sphere 

  • hemisphere

Q 48 | Page 91

If two solid-hemisphere s of same base radius r are joined together along their bases , then curved surface area  of this new solid is 

  • \[4 \pi r^2\]

  • \[6 \pi r^2\]

  • \[3 \pi r^2\]

  • \[8 \pi r^2\]

Q 49 | Page 91

The diameters of two circular ends of the bucket are 44 cm and 24 cm . The height of the bucket is 35 cm . The capacity of the bucket is

  • 32.7 litres 

  •   33.7 litres 

  • 34.7 litres  

  •  31.7 litres 

Q 50 | Page 91

No Question.

Chapter 14: Surface Areas and Volumes

Ex. 14.10Ex. 14.20Ex. 14.30Others

RD Sharma 10 Mathematics

10 Mathematics - Shaalaa.com

RD Sharma solutions for Class 10 Mathematics chapter 14 - Surface Areas and Volumes

RD Sharma solutions for Class 10 Maths chapter 14 (Surface Areas and Volumes) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 14 Surface Areas and Volumes are Introduction of Surface Areas and Volumes, Surface Area of a Combination of Solids, Volume of a Combination of Solids, Conversion of Solid from One Shape to Another, Frustum of a Cone, Surface Areas and Volumes Examples and Solutions.

Using RD Sharma Class 10 solutions Surface Areas and Volumes exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.

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