NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 12: Surface Areas & Volumes

Two identical solid hemispheres of equal base radius \(r\) cm are stuck together along their bases. The total surface area of the combination is \(6\pi r^2\).

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 \(4\pi rh + 4\pi r^2\).

A solid cone of radius \(r\) and height \(h\) is placed over a solid cylinder having same base radius and height as that of the cone. The total surface area of the combined solid is \(\pi r[\sqrt{r^2+h^2}+3r+2h]\).

A solid ball is exactly fitted inside the cubical box of side \(a\). The volume of the ball is \(\dfrac{4}{3}\pi a^3\).

The volume of the frustum of a cone is \(\dfrac{1}{3}\pi h[r_1^2+r_2^2-r_1r_2]\), where \(h\) is vertical height of the frustum and \(r_1, r_2\) are the radii of the ends.

The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom (see Fig. 12.7) is \(\dfrac{\pi r^2}{3}[3h-2r]\).

The curved surface area of a frustum of a cone is \(\pi l(r_1+r_2)\), where \(l=\sqrt{h^2+(r_1+r_2)^2}\), \(r_1,r_2\) are radii and \(h\) is height.

An open metallic bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base. The surface area of the metallic sheet used is equal to curved surface area of frustum + area of circular base + curved surface area of cylinder.

Three metallic solid cubes whose edges are 3 cm, 4 cm and 5 cm are melted and formed into a single cube. Find the edge of the cube so formed.

How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm \(\times\) 11 cm \(\times\) 12 cm?

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.

A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of the two parts.

Two identical cubes each of volume \(64\,\text{cm}^3\) are joined together end to end. What is the surface area of the resulting cuboid?

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.

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.

Two solid cones \(A\) and \(B\) are placed in a cylindrical tube as shown.

The ratio of their capacities is \(2:1\). Find the heights and capacities of the cones. Also find the volume of the remaining portion of the cylinder (tube length 21 cm, inner diameter 6 cm).

An ice-cream cone with hemispherical top has radius 5 cm and height 10 cm (see figure).

Calculate the volume of ice cream, if \(\dfrac16\) of the cone part is left unfilled.

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

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

A wall 24 m long, 0.4 m thick and 6 m high is constructed with bricks each of dimensions 25 cm \(\times\) 16 cm \(\times\) 10 cm. If the mortar occupies \(\dfrac{1}{10}\) of the volume of the wall, find the number of bricks used.

Find the number of metallic circular discs with 1.5 cm base diameter and height 0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

A solid metallic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. Determine the height of the cone.

A rectangular water tank of base \(11\,\text{m}\times 6\,\text{m}\) contains water up to a height of 5 m. If the water is transferred to a cylindrical tank of radius 3.5 m, find the height of water in the cylinder.

How many cubic centimetres of iron are required to construct an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm, the thickness being 1.5 cm? If 1 cm³ of iron weighs 7.5 g, find the weight of the box.

A fountain-pen barrel is a cylinder of length 7 cm and diameter 5 mm. A full barrel writes 3300 words on average. How many words can be written with a bottle containing \(\dfrac15\) litre of ink?

Water flows at \(10\,\text{m min}^{-1}\) through a cylindrical pipe of diameter 5 mm. How long to fill a conical vessel of diameter 40 cm and depth 24 cm?

A heap of rice is a cone of diameter 9 m and height 3.5 m. Find the volume of rice and the canvas required to just cover it.

A factory makes 1,20,000 pencils daily. Each pencil is a cylinder of length 25 cm and base circumference 1.5 cm. Find the cost of colouring the curved surfaces at Rs 0.05 per dm².

Water flows at 15 km/h through a pipe of diameter 14 cm into a cuboidal pond \(50\,\text{m}\times44\,\text{m}\). In what time will the water level rise by 21 cm?

A solid iron cuboid \(4.4\,\text{m}\times2.6\,\text{m}\times1\,\text{m}\) is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.

500 persons take a dip in a cuboidal pond \(80\,\text{m}\times50\,\text{m}\). If the average water displacement per person is \(0.04\,\text{m}^3\), find the rise in water level.

Sixteen glass spheres, each of radius 2 cm, are packed into a cuboidal box of internal dimensions \(16\,\text{cm}\times8\,\text{cm}\times8\,\text{cm}\) and the box is then filled with water. Find the volume of water filled.

A milk container of height 16 cm is a frustum with radii 8 cm and 20 cm at the ends. Find the capacity and the cost of milk at Rs 22 per litre that it can hold.

A cylindrical bucket (height 32 cm, base radius 18 cm) is filled with sand and emptied to form a conical heap of height 24 cm. Find the radius and slant height of the heap.

A rocket is a cylinder (radius 3 cm, height 12 cm) surmounted by a cone of the same radius and slant height 5 cm. Find the total surface area and volume. [Use \(\pi=3.14\)].

A building is a cylinder surmounted by a hemispherical dome and contains \(41\dfrac{19}{21}\,\text{m}^3\) of air. If the internal diameter of the dome equals the total height above the floor, find the height of the building.

A hemispherical bowl of internal radius 9 cm is full of liquid. It is to be filled into cylindrical bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed?

A solid cone (height 120 cm, radius 60 cm) is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. The cylinder radius equals that of the cone. Find the volume of water left in the cylinder.

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

Rain from a roof \(22\,\text{m}\times20\,\text{m}\) drains into a cylindrical vessel of diameter 2 m and height 3.5 m. If the vessel is just filled, find the rainfall in cm.

A wooden pen stand is a cuboid \(10\times5\times4\,\text{cm}\) with four conical depressions (radius 0.5 cm, depth 2.1 cm) and a cubical depression of edge 3 cm. Find the volume of wood in the stand.