R.D. Sharma solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० chapter 14 - Surface Areas and Volumes [Latest edition]

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

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?

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

Find the number of metallic circular discs with a 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.

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

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.

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

How many coins 1.75 cm in diameter and 2 mm thick must be melted to form a cuboid 11 cm × 10 cm × 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 it =`22/7`)

A solid cuboid of iron with dimensions 33 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.

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. 

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.

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 2/5 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?

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 .

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)`

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

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

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

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 

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  ?

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?

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 ? 

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?

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

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.

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 metal used in making cylinder is 176 cm³, find the outer and inner diameters of the cylinder. (Use π = 22/7)

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.

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)

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.

A vessel is in the form of a hollow hemisphere surmounted 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.

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`]

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)

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.

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 the hemisphere, calculate the height of the cone and the surface area of the toy. `("Use"  pi = 22/7)`.

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).

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

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]

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

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. 

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 .

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.

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.

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 so formed.

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.

The difference between outside and inside surface areas of cylindrical metallic pipe 14 cm long is 44 cm². If the pipe is made of 99 cm³ of metal, find the outer and inner radii of the pipe.

The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. If the volume of the metal used in making the cylinder is 176 cm³, find the outer and inner radii of the cylinder.

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 cm^{2 .}[Use`pi=22/7`]

A wall 24 m long, 0.4 m thick and 6 m high is constructed with the bricks each of dimensions 25 cm × 16 cm × 10 cm. If the mortar occupies `1/10` th of the volume of the wall, then find the number of bricks used in constructing the wall.

A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains `41 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?

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains `1408/21 m^3` of air, find the height of the cylindrical part. `("Use"  pi = 22/7)`

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  \[67\frac{1}{21} m^3\].

Two solid cones A 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.

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

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`]

An empty cone of radius 3 cm and height 12 cm is filled with ice-cream such that the lower part of the cone which is `(1/6)^"th"` of the volume of the cone is unfilled (empty) but a hemisphere is formed on the top. Find the volume of the ice-cream.

A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.

If the radii of the bases of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3, find the ratio of their volumes.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. Also, find the surface area of the toy. (Take 𝜋 = 3⋅14)

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)

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 \[\pi = 3 . 14) .\]

A bucket is in the form of a frustum of a cone with a capacity of 12317.6 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).

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?

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] 

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?

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.

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`)

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.

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)

A solid metallic sphere of radius 3 cm is melted and recast into the shape of a solid cylinder of radius 2 cm. Find the height of the cylinder.

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

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

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

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 \[\frac{H}{R} .\]

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.

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

A solid piece of metal in the form of a cuboid of dimensions 11 cm × 7 cm × 7 cm is melted to form 'n' number of solid spheres of radii `7/2` cm each. Find the value of n.

A funnel is the combination of a ______ of a cone and ______.

A cylindrical pencil sharpened at one edge is the combination of a ______ and a ______.

A surahi is the combination of a ______ and a ______.

A plumbline (sahul) is the combination of a ______ and a ______.

The shape of a gilli, in the gilli-danda game, is a combination of ______ and a ______.

A shuttle cock used for playing badminton has the shape of the combination of ______ and ______.

Two identical solid cubes of side a are joined end to end. The total surface area of the resulting cuboid is ______.

If a solid cone of base radius r and height h is placed over a solid cylinder having the same base radius and height as that of the cone, then the curved surface area of the solid so formed is ______.

A solid ball is exactly fitted inside a cuboidal box of side a. The volume of the ball is ______.

A solid cylinder of radius r and height h is placed over another cylinder of same height and radius. The total surface area of the shape so formed is ______.

Two identical solid hemispheres of equal base radius r cm are stuck together along their bases. The total surface area of the combination is ______.

The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom, as shown in figure, is ______.

If a marble of radius 2.1 cm is put into a cylindrical cup, full of water, of radius 5 cm and height 6 cm, then the volume of water that flows out of the cup is ______.

Three solid cubes have a face diagonal of `4sqrt(2)` cm each. Three other solid cubes have a face diagonal of  `8sqrt(2)` cm each. All the cubes are melted together to form a cube. The side of the cube so formed is of length ______.

In figure, a circle is inscribed in a square ABCD and the square is circumscribed by a circle. If the radius of the smaller circle is r cm, then the area of the shaded region in cm² is ______.

The volume of the greatest right circular cone, which can be cut from a cube of side 4 cm is ______.

The volume of a cuboid is `24sqrt(42)` cm³. Its length is `5sqrt(2)` cm, breadth and height are in the ratio `sqrt(3) : sqrt(7)`. The height of the cuboid is ______.

The sum of the length, breadth and height of a cuboid is `5sqrt(3)` cm and length of its diagonal is `3sqrt(5)` cm, then its total surface area is ______.