Class 10 - Arihant - All in one (Math) - Surface Areas and VolumesContact Number: 9667591930 / 8527521718

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Three cubes each side 5 cm are joined end to end. Find the surface area of the resulting solid.

An iron pole consists of a cylinder of height 240 cm and base diameter 26 cm, which is surmounted by another cylinder of height 66 cm and radius 10 cm. Find the mass of the pole given that 1 ${\mathrm{cm}}^{3}$ of iron has approximately 8 g mass. $\left[\mathrm{Take},\mathrm{\pi}=3.14\right]$

The decorative block shown in the following figure is made of two solids, a cube, and a hemisphere. The base of the block is a cube with an edge 5 cm and the hemisphere fixed on the top has a diameter of 4.2 cm. then find the total Surface area of the block and find the total area to be painted. $\left[\mathrm{Take},\mathrm{\pi}=\frac{22}{7}\right]$

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 hemispherical end is 7 cm, then find the cost of polishing its surface at the rate of ₹ 2 per ${\mathrm{dm}}^{2}.\left[\mathrm{Take},\mathrm{\pi}=\frac{22}{7}\right]$

A wooden toy rocket is in the shape of a cone mounted on a cylinder, as shown in the figure. The height of the entire rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 cm. If the conical portion is to be painted orange and the cylindrical portion yellow, then find the area of the rocket painted with each of these colours. [take, $\mathrm{\pi}$ 3.14]

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 4cm. Determine the volume of the solid toy. If a right circular cylinder circumscribes the toy, then find the difference of the volumes oi the cylinder and the toy. [take, $\mathrm{\pi}$ = 3.14]

A solid ball exactly filled inside the cubical box of side a. What is the volume of the remaining space inside the cubical box?

What will be the approximate volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm?

From a circular cylinder of a diameter 10 cm and height 12 cm, a conical cavity of the same base radius and of the same height is hollowed out. Find the volume of the remaining solid. $\left[\mathrm{take},\mathrm{\pi}=3.14\right]$

A metal container is in the form of a cylinder surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7 m and the internal radius of the cylinder is 3.5 m. Calculate the total surface area of the container.

A wooden article was made by scooping out a hemisphere from one face of a cubical wooden block. If each edge of the cube is 10 cm and the diameter of the base of the hemisphere is 7 cm, then find the volume of the wooden article.

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter of 4 units of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

A spherical glass vessel has a cylindrical neck 7 cm long, 4 cm in diameter, the diameter of the spherical part is 21 cm. Find the quantity of water it can hold. $\left(\mathrm{use},\mathrm{\pi}=\frac{22}{7}\right)$

A tent is in the shape of a right circular cylinder up to a height of 3 m and conical above it. The total height of the tent is 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of 2 per m2, if the radius of the base is 14 cm.

A pen stand made of wood is in the shape of a cuboid with four conical depressions and a cubical depression to hold the pens and pins, respectively. The dimensions of the cuboid are 10 cm, 5 cm, and 4 cm. The radius of each of the conical depressions 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 building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41\frac{19}{21}{\mathrm{m}}^{3}$ of air. If the internal diameter of the dome is equal to its total height above the floor. Find the height of the building.

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

A well of diameter 10 m is dug 14 m deep. The Earth taken out of it is spread evenly all around to a width of 5 m 10 form an embankment. Find the height of the embankment.

Water is flowing at the rate Of 5 km/h through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Determine the time in which the level of the water in the tank will rise by 7 cm.

A metallic sphere of radius 10.5 cm is melted and then recast into smaller cones, each of radius 3.5 cm and height 3 cm. How many cones are obtained?

Find the number of cubes of side 2 cm which can be cut of side 6 cm.

Three metallic solid cubes whose edges are 12 cm, 16 cm, and 20 cm respectively, are melted and formed into a single cube. Find the edge of the cube so formed.

If a metallic cube of edge 1 cm is drawn into a wire of diameter 4 mm, then find the length of the wire.

Eight solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 6 cm and height 32 cm. Then, find the diameter of each sphere.

How many spherical lead shots each 4.2 cm in diameter can be obtained from a rectangular solid of lead with dimensions 66 cm, 42 cm, and 21 cm?

A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is filled into cylindrical-shaped bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?

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

Water flows through a circular pipe whose internal diameter is 2 cm at the rate of 0.7 m/s into a cylindrical tank, the radius of whose base is 40 cm. By how much will the level of water rise in the tank in half an hour?

A sphere of diameter 12 cm is dropped into a right circular cylindrical vessel partly filled with water. If the sphere is completely submerged in water, then the water level in the vessel rises by $3\frac{5}{9}$ cm. Find the diameter of the cylindrical vessel.

A well whose diameter is 7 m, has been dug 22.5 m deep and the Earth dugout is used to form an embankment around it. If the height of the embankment is 1.5 m, then find the width of the embankment.

If the radii oi the circular end of a frustum oi height 6 cm is 15 cm, and 7 cm respectively, then find the volume and lateral surface area (curved surface area) oi the frustum. $\left[\mathrm{take},\mathrm{\pi}=3.14\right]$

bucket of height 16 cm and made up of metal sheet is in the form of a frustum of a right circular cone with radii of its lower and upper ends as 3 cm and 15 cm, respectively. Calculate

(i) the height of the cone of which the bucket is a part.

(ii) the volume of water which can be filled in the bucket.

(iii) the slant height of the bucket.

(iv) the area of the metal sheet required to make the bucket.

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 is $\frac{1}{27}$ of the volume of a given cone, then at what height above the base is the section made?

Hanumappa and his wife Gangamma are busy making jaggery out of sugarcane juice. They have processed the sugarcane juice to make the molasses, which is poured into moulds in the shape of a frustum of a cone having the diameters of its two circular face as 30 cm, 35 cm and the vertical height of the mould is 14 cm (see the figure). If 1 ${\mathrm{cm}}^{3}$ of molasses has a mass about 1.2 g, then find the mass of the molasses that can be poured into each mould. $\left[\mathrm{take},\mathrm{\pi}=\frac{22}{7}\right]$

If the radii of circular ends of a frustum of a cone are 20 cm and 12 cm and its height is 6 cm, then find the slant height of frustum (in cm).

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

A frustum of a cone of height 7 cm. The radius of its two circular ends is 6 cm and 3 cm. Find the volume of the frustum.

An open container made up of a metal sheet in the form of the frustum of a cone of height 8 cm with radii of its lower and upper ends as 4 cm and 10 cm, respectively Find the cost of oil which can completely filled the container at the rate of 50 per L. Also, find the cost of metal used, if it costs 50 per 100 ${\mathrm{cm}}^{2}.$

2 cubes of volume 64 ${\mathrm{cm}}^{3}$ are joined end-to-end. find the surface area of the resulting cuboid.

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.

A toy is in the form of a cone of radius 3.5 cm surmounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter of the hemisphere can have? Find the surface area of the solid.

A hemispherical depression is a cut-out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see below figure). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m, respectively and the slant height of the top is 2.8 m, then find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of ₹500 per m. [Note The base of the tent will not be covered with canvas.]

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaming solid to the nearest ${\mathrm{cm}}^{2}.$

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the adjacent figure. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, then find the total surface area of the article.

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\mathrm{\pi}.$

Rachel, an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm.

If each cone has a height of 2 cm, then find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)

A Gulab jamun contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see figure).

A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm x 10 cm x 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see figure).

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 iron pole consists of a cylinder of height 220 cm and a base diameter of 24 cm. which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass Of the pole, given that 1 ${\mathrm{cm}}^{3}$ of iron has approximately 8 g mass. $\left[\mathrm{take},\mathrm{\pi}=3.14\right]$

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.

A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 ${\mathrm{cm}}^{3}.$ Check whether she is correct, taking the above as the inside measurements and $\mathrm{\pi}$ = 3.14.

A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

Metallic spheres of radii 6 cm, 8 cm, and 10 cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

A 20 m deep well with diameter 7 m is dug and the Earth from digging is evenly spread out to form a platform 22 m x 14 m. Find the height of the platform.

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.

A container shaped like a right circular cylinder having a diameter of 12 cm and a height of 15 cm is full of ice cream. The ice-cream is to be filled into 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.

How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm x 10 cm x 3.5 cm?

A cylindrical bucket, 32 Cm high and with the 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, then find the radius and slant height of the heap.

Water in a canal, 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h. How much area will it irrigate in 30 min, if 8 cm of standing water is needed?

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?

A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.

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.

A fez, the cap used by the Turks, is shaped like the frustum of a cone (see the figure). If its radius on the open side is 10 cm, the radius at the upper base is 4 cm and its slant height is 15 cm, then find the area of material used for making it.

A container opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container at the rate of ₹ 20 per L. Also, find the cost of metal sheet used to make the container, if it costs ₹ 8 per 100 ${\mathrm{cm}}^{2}.$ [take, $\mathrm{\pi}$ = 3.14]

A metallic right circular cone 20 cm high and whose vertical angle is $60\xb0,$ is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{16}$ cm, then find the length of the wire.

Copper wire 3 mm in diameter is wound about a cylinder whose length is 12 cm and diameter 10 cm, so as to cover the curved surface Of the cylinder. Find the length and mass of the wire, assuming the density of copper to be 8.88 g per ${\mathrm{cm}}^{3}.$

A right triangle whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. [Choose the value of as found appropriate]

A cistern, internally measuring 150 cm x 120 cm x 110 cm. has 129600 ${\mathrm{cm}}^{3}$ of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each brick being 225 cm x 7S cm x 65 cm?

An oil funnel made of the tin sheet consists of d 10 cm long cylindrical portion attached to a frustum of a cone. If the total height is 22 cm, the diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, then find the area of the tin sheet required to make the funnel.

Derive the formula for the curved surface area and total surface area of the frustum of a cone (using the symbols as explained).

Derive the formula for the volume of the frustum of a cone (using the symbols as explained).

A spherical ball of diameter 21 cm is melted and recast into cubes, each of sides 1 cm. Find the number of cubes so formed.

Find the ratio of the volumes of a cube to that of a sphere, which will exactly fit inside the cube.

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

If the radii of circular ends of the frustum of a cone are 17 cm and 15 cm and its height is 4 cm, then find the slant height of frustum (in cm).

Volumes of two spheres are in the ratio 64:27. What will be the ratio of their surface area?

A cone is divided into two equal parts by drawing a plane through the mid-point of its axis, parallel to its base. Find the ratio of volumes of two parts.

A solid of iron in the form of a cuboid of dimension 49 cm x 33 cm x 24 cm is molded to form a solid sphere. Find the radius of the sphere.

Find the volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm.

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

From a circular cylinder of diameter, 10 cm and height 12 cm. a conical cavity the same base radius and o! the same height is hollowed out. Find the volume of the remaining solid. [take, $\mathrm{\pi}$ = 314]

50 circular plates, each of radius 7 cm and thickness 0.5 cm. are placed one above another to form a solid right circular cylinder Find the total surface area and the volume of the cylinder so formed.

A solid cuboidal slab of iron of dimensions 66 cm x 20 cm X 27 cm is used to cast an iron pipe. If the outer diameter of the pipe is 10 cm and the thickness is 1 cm. then calculate the length of the pipe.

how many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm x 11 cm x 12 cm?

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.

Water flows through a cylindrical pipe, whose inner radius is 1 cm, at the rate of 80 cm/s in an empty cylindrical tank, the radius of whose base is 40 cm. What is the rise of the water level in the tank in half an hour?

A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8 cm. Find the height of the cone.

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

An hourglass is made using identical double glass cones of diameter 10 cm each. If the total height is 24 cm, then find the surface area of the glass used in making it.

A warehouse is used as a granary. It is in the shape of a cuboid surmounted by a half-cylinder. The base of the warehouse is 6 m x 14 m and its height is 8 m. Find the surface area of the non-cuboidal part of the warehouse.

A small terrace at a hockey ground comprises of 10 steps each of which 20m long and built of solid concrete. Each step has a rise of $\frac{1}{4}\mathrm{m}$ and a tread of $\frac{1}{2}\mathrm{m}.$ Calculate the total volume of concrete required to build the terrace.

A solid metallic cylinder of radius 3.5 cm height 14 cm is melted and recast into a number of small solid metallic balls, each of radius $\frac{7}{12}\mathrm{cm}.$ Find the number of balls so formed.

The cost of the painting of the total outside surface Of a closed cylindrical oil tank at 60 paise per sq m is 237.60 and the height of the tank is 6 times the radius of the base of the tank. Find the radius and tank.[take, $\mathrm{\pi}$ = 22/7]

A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of the cylinder. The diameter and height of the cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, find the total surface area and volume of the rocket. [take, $\mathrm{\pi}$ = 3.14]

A circus tent is made up of using two different colored cloth material. Red-colored material is used to make cylindrical parts up to a height of 3 m and green colored material to make conical part above it. It the diameter of the base is 105 m and the slant height of the conical part is 53 m, find the red colored material and green coloured material required. [assuming on stitching margins.]

A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that $\frac{1}{8}$ space of the cube remains unfilled. Then, find the number of marbles that the cube can accommodate.

A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm. respectively. Find the cost of the bucket, if the cost of metal sheet used is ₹ 15 per 100 ${\mathrm{cm}}^{2}.$ [take, $\mathrm{\pi}$ = 3.14]

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 solid right circular Cones nave the same height. The radii of their bases are ${\mathrm{r}}_{1}$ and ${\mathrm{r}}_{2}$ They are melted and recast into a cylinder of the same height. Show that the radius of the base of the cylinder is $\sqrt{\frac{{\mathrm{r}}_{1}^{2}+{\mathrm{r}}_{2}^{2}}{3}}$.

A conical vessel with base radius 5 cm and height 24 cm, is full 01 water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. $\left[\mathrm{take},\mathrm{\pi}=\frac{22}{7}\right]$

The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 0.5 cm in diameter. A full barrel of ink in the pen can be used for writing 275 words on average. How many words would be written using a bottle of ink containing one-fourth of a liter?

A trophy awarded to the best student in the class is in the form of a solid cylinder mounted on a solid hemisphere with the same radius and is made from some metal. This trophy is mounted on a wooden cuboid as shown in the figure.

The diameter of the hemisphere is 21 cm and the total height of the trophy is 24.5 cm. Find the weight of the metal used in making the trophy, if the weight of 1 cm of the metal is 1.2 g. $\left[\mathrm{take},\mathrm{\pi}=\frac{22}{7}\right]$

The inner diameter of a cylindrical container is 7 cm and its top is of the shape of a hemisphere. If the height of the container is 16 cm, then find the actual capacity of the container. $\left[\mathrm{take},\mathrm{\pi}=\frac{22}{7}\right]$

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the 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 between the volumes of cube and the toy. Also, find the total surface area of the toy.

A building is in the form of a cylinder surmounted by a hemispherical dome (see the figure). 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 $67\frac{1}{21}{\mathrm{m}}^{3}$ of air.

The interior of a building is in the form of a cylinder of diameter 4.3 m and 3.8 m height, surmounted by a cone whose vertical angle is a right angle. Find the area of the surface and the volume of the building.

A farmer wants to dig a well either in the form of cuboid dimensions (1 m xl m x 7 m) or in the form of the cylinder of diameter 1m and height 7 m. The rate to dig the well is ₹ 50 per ${\mathrm{m}}^{3}.$

(i) Find the cost to dig both wells.

(ii) The farmer decides to dig the cylindrical well. by his decision which value is depicted here?

Due to heavy floods in a state, thousands were rendered homeless. 50 schools collectively offered to the state government to provide the place and the canvas 1500 tents to be tired by the government and decided to share their whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 m and height 3 m, with the conical upper part of the same base radius but of height 2.1 m. If the canvas used to make the tents cost ₹ 120 per ${\mathrm{m}}^{2}$. then tind the amount shared by to set up the tents. What value is generated by the above problem? $\left[\mathrm{take},\mathrm{\pi}=\frac{22}{7}\right]$

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