Class 10 - Chapter 12: Surface Areas and Volumes

Exercise 12.2

Q1. 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 π.

Given:

Radius of hemisphere (r) = 1 cm

Radius of cone (r) = 1 cm

Height of cone (h) = 1 cm

Volume of Hemisphere

Volume = (2/3)πr³

= (2/3)π(1)³

= (2π/3) cm³

Volume of Cone

Volume = (1/3)πr²h

= (1/3)π(1)²(1)

= (π/3) cm³

Total Volume of Solid

= Volume of Hemisphere + Volume of Cone

= (2π/3) + (π/3)

= π cm³

Volume of the solid = π cm³

Q2. 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.

Given:

Diameter = 3 cm

Radius (r) = 1.5 cm

Total length of model = 12 cm

Height of each cone = 2 cm

Height of Cylinder

Cylinder height

= 12 − 2 − 2

= 8 cm

Volume of Cylinder

V₁ = πr²h

= (22/7) × (1.5)² × 8

= (22/7) × 2.25 × 8

= 396/7 cm³

Volume of Two Cones

Volume of one cone

= (1/3)πr²h

= (1/3) × (22/7) × (1.5)² × 2

= 33/7 cm³

Volume of two cones

= 66/7 cm³

Total Volume of Air

= Volume of cylinder + Volume of two cones

= 396/7 + 66/7

= 462/7

= 66 cm³

Volume of air contained in the model = 66 cm³

Q3. 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.

Given:

Length of gulab jamun = 5 cm

Diameter = 2.8 cm

Radius (r) = 1.4 cm

Length of cylindrical part

= 5 − 2r

= 5 − 2.8

= 2.2 cm

Volume of one gulab jamun

= Volume of cylinder + Volume of sphere

= πr²h + (4/3)πr³

= (22/7)(1.4)²(2.2) + (4/3)(22/7)(1.4)³

= 13.552 + 11.499

≈ 25.051 cm³

Syrup in one gulab jamun

= 30% of 25.051

≈ 7.515 cm³

Syrup in 45 gulab jamuns

= 45 × 7.515

≈ 338.18 cm³

Amount of syrup ≈ 338 cm³

Q4. 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 depression is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.

Volume of Cuboid

= l × b × h

= 15 × 10 × 3.5

= 525 cm³

Volume of one conical depression

= (1/3)πr²h

= (1/3)(22/7)(0.5)²(1.4)

= 11/30 cm³

Volume of four depressions

= 4 × 11/30

= 22/15 cm³

Volume of wood

= 525 − 22/15

= 7853/15

≈ 523.53 cm³

Volume of wood ≈ 523.53 cm³

Q5. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top 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 into the vessel.

Volume of cone

= (1/3)πr²h

= (1/3)(22/7)(5²)(8)

= 4400/21 cm³

Water overflowed

= (1/4) × 4400/21

= 1100/21 cm³

Volume of one lead shot

= (4/3)πr³

= (4/3)(22/7)(0.5)³

= 11/21 cm³

Number of lead shots

= (1100/21) ÷ (11/21)

= 100

Number of lead shots = 100

Q6. 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. (Use π = 3.14)

Lower Cylinder

Radius = 12 cm

Volume = 3.14 × 12² × 220

= 99,475.2 cm³

Upper Cylinder

Radius = 8 cm

Volume = 3.14 × 8² × 60

= 12,057.6 cm³

Total Volume

= 99,475.2 + 12,057.6

= 111,532.8 cm³

Mass

= 111,532.8 × 8

= 892,262.4 g

= 892.26 kg

Mass of the pole ≈ 892.26 kg

Q7. 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.

Volume of Cylinder

= πr²h

= (22/7)(60²)(180)

= 2,036,571.43 cm³

Volume of Cone

= (1/3)πr²h

= (1/3)(22/7)(60²)(120)

= 452,571.43 cm³

Volume of Hemisphere

= (2/3)πr³

= (2/3)(22/7)(60³)

= 452,571.43 cm³

Total Volume of Solid

= 905,142.86 cm³

Water Left

= 2,036,571.43 − 905,142.86

= 1,131,428.57 cm³

Volume of water left ≈ 1,131,428.57 cm³

Q8. 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. (Take π = 3.14)

Radius of spherical part

= 8.5/2

= 4.25 cm

Radius of neck

= 1 cm

Height of neck

= 8 cm

Volume of spherical part

= (4/3)πr³

= (4/3)(3.14)(4.25)³

≈ 321.39 cm³

Volume of cylindrical neck

= πr²h

= 3.14 × 1² × 8

= 25.12 cm³

Total Volume

= 321.39 + 25.12

= 346.51 cm³

This is approximately 345 cm³.

Yes, the child's measurement is correct (approximately).