Chapter 12: Surface Areas and Volumes

Q1Written answer

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 being equal to its radius. Find the volume of the solid in terms of π.

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Worked solution

Here r = 1 cm and h = 1 cm.

The diagram is as follows.

Ncert solutions class 10 chapter 13-12

Now, Volume of solid = Volume of conical part + Volume of hemispherical part

We know the volume of cone = ⅓ πr²h

And,

The volume of the hemisphere = ⅔πr³

So, the volume of the solid will be

Ncert solutions class 10 chapter 13-13

= π cm³

Q2Written answer

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, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model are nearly the same.)

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Worked solution

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.2 Solution 2

Given,

Height of cylinder = 12–4 = 8 cm

Radius = 1.5 cm

Height of cone = 2 cm

Now, the total volume of the air contained will be = Volume of cylinder+2×(Volume of the cone)

∴ Total volume = πr²h+[2×(⅓ πr²h )]

= 18 π+2(1.5 π)

= 66 cm³.

Q3Written answer

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 a length of 5 cm and a diameter of 2.8 cm (see figure).

Ncert solutions class 10 chapter 13-14

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Worked solution

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.2 Solution 3

It is known that the gulab jamuns are similar to a cylinder with two hemispherical ends.

So, the total height of a gulab jamun = 5 cm.

Diameter = 2.8 cm

So, radius = 1.4 cm

∴ The height of the cylindrical part = 5 cm–(1.4+1.4) cm

=2.2 cm

Now, the total volume of one gulab jamun = Volume of cylinder + Volume of two hemispheres

= πr²h+(4/3)πr³

= 4.312π+(10.976/3) π

= 25.05 cm³

We know that the volume of sugar syrup = 30% of the total volume

So, the volume of sugar syrup in 45 gulab jamuns = 45×30%(25.05 cm³)

= 45×7.515 = 338.184 cm³

Q4Written answer

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 by 10 cm by 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 Fig.).

Ncert solutions class 10 chapter 13-15

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Worked solution

The volume of the cuboid = length x width x height

We know the cuboid’s dimensions as 15 cmx10 cmx3.5 cm

So, the volume of the cuboid = 15x10x3.5 = 525 cm³

Here, depressions are like cones, and we know,

Volume of cone = (⅓)πr²h

Given, radius (r) = 0.5 cm and depth (h) = 1.4 cm

∴ Volume of 4 cones = 4x(⅓)πr²h

= 1.46 cm²

Now, the volume of wood = Volume of the cuboid – 4 x volume of the cone

= 525-1.46 = 523.54 cm²

Q5Written answer

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.

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Worked solution

For the cone,

Radius = 5 cm,

Height = 8 cm

Also,

Radius of sphere = 0.5 cm

The diagram will be like

Ncert solutions class 10 chapter 13-16

It is known that,

The volume of cone = volume of water in the cone

= ⅓πr²h = (200/3)π cm³

Now,

Total volume of water overflown= (¼)×(200/3) π =(50/3)π

The volume of lead shot

= (4/3)πr³

= (1/6) π

Now,

The number of lead shots = Total volume of water overflown/Volume of lead shot

= (50/3)π/(⅙)π

= (50/3)×6 = 100

Q6Written answer

A solid iron pole consists of a cylinder of height 220 cm and base diameter 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 cm³ of iron has approximately 8 g mass.

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Worked solution

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.2 Solution 6

Given the height of the big cylinder (H) = 220 cm

The radius of the base (R) = 24/2 = 12 cm

So, the volume of the big cylinder = πR²H

= π(12)²× 220 cm³

= 99565.8 cm³

Now, the height of the smaller cylinder (h) = 60 cm

The radius of the base (r) = 8 cm

So, the volume of the smaller cylinder = πr²h

= π(8)²×60 cm³

= 12068.5 cm³

∴ The volume of iron = Volume of the big cylinder+ Volume of the small cylinder

= 99565.8 + 12068.5

=111634.5 cm³

We know,

Mass = Density x volume

So, the mass of the pole = 8×111634.5

= 893 Kg (approx.)

Q7Written answer

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.

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Worked solution

Here, the volume of water left will be = Volume of the cylinder – Volume of solid

Given,

Radius of cone = 60 cm,

Height of cone = 120 cm

Radius of cylinder = 60 cm

Height of cylinder = 180 cm

Radius of hemisphere = 60 cm

Now,

The total volume of solid = Volume of Cone + Volume of the hemisphere

Volume of cone = 1/3πr²h = 1/3 × π×60²×120cm³ = 144×10³π cm³

Volume of hemisphere = (⅔)×π×60³cm³ = 144×10³π cm³

So, total volume of solid = 144×10³π cm³ + 144×10³π cm³ = 288 ×10³π cm³

Volume of cylinder = π×60²×180 = 648000 = 648×10³ π cm³

Now, the volume of water left will be = Volume of the cylinder – Volume of solid

= (648-288) × 10³×π = 1.131 m³

Q8Written answer

A spherical glass vessel has a cylindrical neck 8 cm long and 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 cm³. Check whether she is correct, taking the above as the inside measurements and π = 3.14.

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Worked solution

Given,

For the cylinder part, Height (h) = 8 cm and Radius (R) = (2/2) cm = 1 cm

For the spherical part, Radius (r) = (8.5/2) = 4.25 cm

Now, volume of this vessel = Volume of cylinder + Volume of sphere

= π×(1)²×8+(4/3)π(4.25)³

= 346.51 cm³

Hence, the child’s calculation is not correct.

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Exercise: 13.2 (