#### Chapters

Chapter 2 - Exponents of Real Numbers

Chapter 3 - Rationalisation

Chapter 4 - Algebraic Identities

Chapter 5 - Factorisation of Algebraic Expressions

Chapter 6 - Factorisation of Polynomials

Chapter 7 - Introduction to Euclid’s Geometry

Chapter 8 - Lines and Angles

Chapter 9 - Triangle and its Angles

Chapter 10 - Congruent Triangles

Chapter 11 - Co-ordinate Geometry

Chapter 12 - Heron’s Formula

Chapter 13 - Linear Equations in Two Variables

Chapter 14 - Quadrilaterals

Chapter 15 - Areas of Parallelograms and Triangles

Chapter 16 - Circles

Chapter 17 - Constructions

Chapter 18 - Surface Areas and Volume of a Cuboid and Cube

Chapter 19 - Surface Areas and Volume of a Circular Cylinder

Chapter 20 - Surface Areas and Volume of A Right Circular Cone

Chapter 21 - Surface Areas and Volume of a Sphere

Chapter 22 - Tabular Representation of Statistical Data

Chapter 23 - Graphical Representation of Statistical Data

Chapter 24 - Measures of Central Tendency

Chapter 25 - Probability

## Chapter 19 - Surface Areas and Volume of a Circular Cylinder

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Find the lateral surface area and total surface area of a cuboid of length 80 cm, breadth 40

cm and height 20 cm.

Find the lateral surface area and total surface area of a cube of edge 10 cm.

Find the ratio of the total surface area and lateral surface area of a cube.

Mary wants to decorate her Christmas tree. She wants to place the tree on a wooden block

covered with coloured paper with picture of Santa Claus on it. She must know the exact

quantity of paper to buy for this purpose. If the box has length, breadth and height as 80

cm, 40 cm and 20 cm respectively. How many square sheets of paper of side 40 cm would

she require?

The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost

of white washing the walls of the room and the ceiling at the rate of Rs. 7.50 m2.

Three equal cubes are placed adjacently in a row. Find the ratio of total surface area of the new cuboid to that of the sum of the surface areas of the three cubes.

A 4 cm cube is cut into 1 cm cubes. Calculate the total surface area of all the small cubes.

The length of a hall is 18 m and the width 12 m. The sum of the areas of the floor and the

flat roof is equal to the sum of the areas of the four walls. Find the height of the hall.

Hameed has built a cubical water tank with lid for his house, with each other edge 1 .5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm. Find how much he would spend for the tiles, if the cost of tiles is Rs. 360 per dozen.

Each edge of a cube is increased by 50%. Find the percentage increase in the surface area of the cube.

The dimensions of a rectangular box are in the ratio of 2 : 3 : 4 and the difference between the cost ofcovering it with sheet of paper at the rates of Rs. 8 and Rs. 9.50 per m2 is Rs.1248. Find the dimensions of the box.

A closed iron tank 12 m long, 9 m wide and 4 m deep is to be made. Determine the cost of iron sheet used at the rate of Rs. 5 per metre sheet, sheet being 2 m wide.

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

base dimensions 4 m × 3m?

An open box is made of wood 3 cm thick. Its external length, breadth and height are 1.48 m, 1.16 m and 8.3 m. Find the cost of painting the inner surface of Rs 50 per sq. metre.

The cost of preparing the walls of a room 12 m long at the rate of Rs. 1.35 per square metre is Rs. 340.20 and the cost of matting the floor at 85 paise per square metre is Rs. 91.80. Find the height of the room.

The dimensions of a room are 12.5 m by 9 m by 7 m. There are 2 doors and 4 windows in the room; each door measures 2.5 m by 1 .2 m and each window 1 .5 m by I m. Find the cost of painting the walls at Rs. 3.50 per square metre.

The length and breadth of a hall are in the ratio 4: 3 and its height is 5.5 metres. The cost of decorating its walls (including doors and windows) at Rs. 6.60 per square metre is Rs. 5082. Find the length and breadth of the room.

A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm, Breadth = 85 cm (See Fig. 18.5). The thickness of the plank is 5 cm everywhere. The external

faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per cm2 and the rate of painting is 10 paise per cm2. Find the total expenses required for polishing and painting the surface of the bookshelf.

The paint in a certain container is sufficient to paint on area equal to 9.375 m2. How manybricks of dimension 22.5 cm × 10 cm × 7.5 cm can be painted out of this container?

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A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold?

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

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

If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that

`1/V=2/S(1/a+1/b+1)`

The areas of three adjacent faces of a cuboid are x, y and z. If the volume is V, prove that

`V^2` = `xyz`

If the area of three adjacent faces of a cuboid are 8 `cm^2`, 18 `cm^3` and 25 `cm^3`. Find the

volume of the cuboid.

The breadth of a room is twice its height, one half of its length and the volume of the room is 512cu. m. Find its dimensions.

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

Water in a canal 30 cm wide and 12 cm deep, is flowing with a velocity of l00 km per hour.How much area will it irrigate in 30 minutes if 8 cm of standing water is desired?

Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the volume, surface area and diagonal of the new cube.

Two cubes, each of volume 512 cm3 are joined end to end. Find the surface area of the resulting cuboid.

Half cubic meter of gold-sheet is extended by hammering so as to cover an area of 1 hectare. Find the thickness of the gold-sheet.

A metal cube of edge 12 cm is melted and formed into three smaller cubes. If the edges of the two smaller cubes are 6 cm and 8 cm, find the edge of the third smaller cube.

The dimensions of a cinema hall are 100 m, 50 m and 18 m. How many persons can sit in the hall, if each person requires 150 `m^3` of air?

Given that 1 cubic cm of marble weighs 0.25 kg, the weight of marble block 28 cm in width and 5 cm thick is 112 kg. Find the length of the block.

A box with lid is made of 2 cm thick wood. Its external length, breadth and height are 25 cm, 18 cm and 15 cm respectively. How much cubic cm of a liquid can be placed in it? Also, find the volume of the wood used in it.

The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm. The box is made of 1.5 cm thick wood. How many bricks of size 6 cm x 3 cm x 0.75 cm can be put in this box?

How many cubic centimeters of iron are there in an open box whose external dimensions are 36 cm, 25 cm and I 6.5 cm, the iron being 1.5 cm thick throughout? If I cubic cm of iron weighs 15g, find the weight of the empty box in kg.

A cube of 9 cm edge is immersed completely in a rectangular vessel containing water. Ifthe dimensions of the base are 15 cm and 12 cm, find the rise in water level in the vessel.

A rectangular container, whose base is a square of side 5 cm, stands on a horizontal table, and holds water up to 1 cm from the top. When a cube is placed in the water it is completely submerged, the water rises to the top and 2 cubic cm of water overflows. Calculate the volume of the cube and also the length of its edge.

A field is 200 m long and 150 m broad. There is a plot, 50 m long and 40 m broad, near thefield. The plot is dug 7 m deep and the earth taken out is spread evenly on the field. Byhow many meters is the level of the field raised? Give the answer to the second place of decimal.

A field is in the form of a rectangle of length 18 m and width 15 m. A pit, 7.5 m long, 6 m broad and 0.8 m deep, is dug in a corner of the field and the earth taken out is spread over the remaining area of the field. Find out the extent to which the level of the field has been raised.

A rectangular tank is 80 m long and 25 m broad. Water flows into it through a pipe whose cross-section is 25 cm2, at the rate of 16 km per hour. How much the level of the water rises in the tank in 45 minutes.

Water in a rectangular reservoir having base 80 m by 60 m i s 6.5 m deep. In what time can the water be emptied by a pipe ôf which the cross-section is a square of side 20 cm, if the water runs through the pipe at the rate of 15 km/hr.

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

A child playing with building blocks, which are of the shape of the cubes, has built a structure as shown in Fig. 18.12 If the edge of each cube is 3 cm, find the volume of the

structure built by the child.

A godown measures 40m × 25 m × 10 m. Find the maximum number of wooden crates each measuring 1.5 m

× 1.25 m × 0.5 m that can be stored in the godown.

A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm × 12 cm × 8 cm, How many bricks would be required?

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Curved surface area of a right circular cylinder is 4.4 `m^2`. If the radius of the base of the cylinder is 0.7 m, find its height.

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

A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of 12.50 per `m^2`.

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

A solid cylinder has total surface area of 462 cm2. Its curved surface area is one-third of its total surface area. Find the radius and height of the cylinder.

The total surface area of a hollow cylinder which is open from both sides is 4620 sq. cm,area of base ring is 115.5 sq. cm and height 7 cm. Find the thickness of the cylinder

Find the ratio between the total surface area of a cylinder to its curved surface area, given that its height and radius are 7.5 cm and 3.5 cm.

The total surface area of a hollow metal cylinder, open at both ends of external radius 8 cm and height 10 cm is 338 p `cm^2`. Taking r to be inner radius, obtain an equation in r and use it to obtain the thickness of the metal in the cylinder.

A cylindrical vessel, without lid, has to be tin-coated on its both sides. If the radius of the base is 70 cm and its height is 1.4 m, calculate the cost of tin-coating at the rate of Rs. 3.50

per 1000 cm2.

The inner diameter of a circular well is 3.5 m. It is 10 m deep Find:

(i) inner curved surface area.

(ii) the cost of plastering this curved surface at the rate of Rs. `40 per m^2`

Find the lateral curved surface area of a cylinderical petrol storage tank that is 4.2 m in diameter and 4.5 m high. How much steel was actually used, if `1/12` of steel actually used was wasted in making the closed tank?

The students of a Vidyalaya were asked to participate in a competition for making and decorating pen holders in the shape of a cylinder with a base, using cardboard. Each pen

holder was to be of radius 3 cm and height 10.5 cm. The Vidyalaya was to supply the competitors with cardboard. If there were 35 competitors, how much cardboard was

required to be bought for the competition?

The diameter of roller 1.5 m long is 84 cm. If it takes 100 revolutions to level a playground, find the cost of levelling this ground at the rate of 50 paise per square metre.

Twenty cylindrical pillars of the Parliament House are to be cleaned. If the diameter of each pillar is 0.50 m and height is 4 m. What will be the cost of cleaning them at the rate of

Rs. 2.50 per square metre?

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Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.

The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.

The radius of a cone is 7 cm and area of curved surface is 176 `cm^2`. Find the slant height.

The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.

Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.

Find the curved surface area of a cone with base radius 5.25 cm and slant height 10cm.

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

The area of the curved surface of a cone is 60 cm2. If the slant height of the cone be 8 cm, find the radius of the base?

The curved surface area of a cone is 4070 cm2 and its diameter is 70 cm. What is its slant height? (Use it 𝜋 = 22/7).

The radius and slant height of a cone are In the ratio of 4 : 7. If its curved surface area is 792 cm2, find its radius. (Use it 𝜋 = 22/7).

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

Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4 : 3.

There are two cones. The curved surface area of one is twice that of the other. The slant height of the later is twice that of the former. Find the ratio of their radii.

The diameters of two cones are equal. If their slant heights are in the ratio 5 : 4, find the ratio of their curved surfaces.

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

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs. 210 per l00 m2.

A conical tent is 10 m high and the radius of its base is 24 m. Find the slant height of the tent. If the cost of 1 2 m canvas is Rs. 70, find the cost of the canvas required to make the tent.

A circus tent is cylindrical to a height of 3 meters 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.

The circumference of the base of a 10 m height conical tent is 44 metres. Calculate the length of canvas used in making the tent if width of canvas is 2 m. (Use it 𝜋= 22/7).

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

A bus stop is barricated from the remaining part of the road, by using 50 hollow cones made of recycled card-board. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs. 12 per m2, what will be the cost of painting all these cones. (Use 𝜋 = 3.14 and √1.04 = 1.02)

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the radius of each is to the height of each as 3:4.

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Find the volume of a right circular cone with:

radius 6 cm, height 7 cm.

Find the volume of a right circular cone with:

radius 3.5 cm, height 12 cm

Find the volume of a right circular cone with:

height 21 cm and slant height 28 cm.

Find the capacity in litres of a conical vessel with radius 7 cm, slant height 25 cm

`["Assume "pi=22/7]`

Find the capacity in litres of a conical vessel with height 12 cm, slant height 13 cm

`["Assume "pi=22/7]`

Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.

The radius and the height of a right circular cone are in the ratio 5 : 12. If its volume is 314 cubic meter, find the slant height and the radius (Use it 𝜋 = 3.14).

The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the slant height and radius of the cone. (Use it 𝜋 = 3.14).

The ratio of volumes of two cones is 4 : 5 and the ratio of the radii of their bases is 2:3. Find the ratio of their vertical heights.

A cylinder and a cone have equal radii of their bases and equal heights. Show that their volumes are in the ratio 3:1.

If the radius of the base of a cone is halved, keeping the height same, what is the ratio of the volume of the reduced cone to that of the original cone?

A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap? (Use 𝜋 = 3.14).

A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap? (Use 𝜋 = 3.14).

A right angled triangle of which the sides containing he right angle are 6.3 cm and lo cm in length, is made to turn round on the longer side. Find the volume of the solid, thus generated. Also, find its curved surface area.

Find the volume of the largest right circular cone that can be fitted in a cube whose edge is 14 cm.

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

(i) height of the cone (ii) slant height of the cone (iii) curved surface area of the cone.

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

Monica has a piece of Canvas whose area is 551 m2. She uses it to have a conical tent made, with a base radius of 7m. Assuming that all the stitching margins and wastage incurred while cutting, amounts to approximately 1 m2. Find the volume of the tent that can be made with it.

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Find the surface area of a sphere of radius 10.5 cm .

Find the surface area of a sphere of radius 5.6 cm .

Find the surface area of a sphere of radius 14 cm .

Find the surface area of a sphere of diameter 14 cm .

Find the surface area of a sphere of diameter 21 cm .

Find the surface area of a sphere of diameter 3.5 cm .

Find the total surface area of a hemisphere and a solid hemisphere each of radius 10 cm.

(Use 𝜋 = 3.14)

The surface area of a sphere is 5544 `cm^2`, find its diameter.

A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin- plating

it on the inside at the rate of Rs. 4 per 100 `cm^2`

The dome of a building is in the form of a hemisphere. Its radius is 63 dm. Find the cost of

painting it at the rate of Rs. 2 per sq. m.

Assuming the earth to be a sphere of radius 6370 km, how many square kilo metres is area

of the land, if three-fourth of the earth’s surface is covered by water?

A cylinder of same height and radius is placed on the top of a hemisphere. Find the curved

surface area of the shape if the length of the shape be 7 cm.

A wooden toy is in the form of a cone surmounted on a hemisphere. The diameter of the base

of the cone is 16 cm and its height is 15 cm. Find the cost of painting the toy at Rs. 7 per 100

`cm^2`.

A storage tank consists of a circular cylinder with a hemisphere adjoined on either end. If the

external diameter of the cylinder be 1.4 m and its length be 8 m, find the cost of painting it

on the outside at the rate of Rs. 10 per `m^2`.

The diameter of the moon is approximately one fourth of the diameter of the earth. Find the

ratio of their surface areas.

A hemi-spherical dome of a building needs to be painted. If the circumference of the base of

the dome is 17.6 cm, find the cost of painting it, given the cost of painting is Rs. 5 per l00

`cm^2`

The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in the given figure. Eight such spheres are used for this purpose, and are to be painted silver. Each support is a cylinder of radius 1.5 cm and height 7 cm and is to be painted black. Find the cost of paint required if silver paint costs 25 paise per cm^{2} and black paint costs 5 paise per cm^{2}.

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Find the volume of a sphere whose radius is 2 cm.

Find the volume of a sphere whose radius is 3.5 cm.

Find the volume of a sphere whose radius is 10.5 cm .

Find the volume of a sphere whose diameter is 14 cm .

Find the volume of a sphere whose diameter is 3.5 dm .

Find the volume of a sphere whose diameter is 2.1 m .

A hemispherical tank has inner radius of 2.8 m. Find its capacity in litres.

A hemispherical bowl is made of steel 0.25 cm thick. The inside radius of the bowl is 5 cm. Find the volume of steel used in making the bowl.

How mañy bullets can be made out of a cube of lead, whose edge measures 22 cm, each bullet being 2 cm in diameter?

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

A spherical ball of lead 3 cm in diameter is melted and recast into three spherical balls. If the diameters of two balls be `a/2`cm and 2 cm, find the diameter of the third ball.

A sphere of radius 5 cm is immersed in water filled in a cylinder, the level of water rises `5/a`cm. Find the radius of the cylinder.

If the radius of a sphere is doubled, what is the ratio of the volume of the first sphere to that of the second sphere?

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

A vessel in the form of a hemispherical bowl is full of water. Its contents are emptied in a right circular cylinder. The internal radii of the bowl and the cylinder are 3.5 cm and 7 cm respectively. Find the height to which the water will rise in the cylinder.

A cylinder whose height is two thirds of its diameter, has the same volume as a sphere of radius 4 cm. Calculate the radius of the base of the cylinder.

A vessel in the form of a hemispherical bowl is full of water. The contents are emptied into a cylinder. The internal radii of the bowl and cylinder are respectively 6 cm and 4 cm. Find the height of water in the cylinder

A cylindrical tub of radius 16 cm contains water to a depth of 30 cm. A spherical iron ball is dropped into the tub and thus level of water is raised by 9 cm. What is the radius of the ball?

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

(Use 𝜋 = 22/7).

The diameter of a coper 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.

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 measuring jar of internal diameter 10 cm is partially filled with water. Four equal spherical balls of diameter 2 cm each are dropped in it and they sink down in water completely. What will be the change in the level of water in the jar?

The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 0.2 cm. Find the length of the wire.

The radius of the internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. If it is melted and recast into a solid cylinder of heigh2`2/a`cm Find the diameter of the cylinder.

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.

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.

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.

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

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.

A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical form ball is dropped into the tub and thus the level of water is raised by 6.75 cm. What is the radius of the ball?

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

A sphere, a cylinder and a cone have the same diameter. The height of the cylinder and also the cone are equal to the diameter of the sphere. Find the ratio of their volumes.

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

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

A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine `("in " mm^3)` is needed to fill this capsule?

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