NCERT Solutions for Class 10 Maths Exercise 12.2 Chapter 12 Surface Areas and Volumes
1. 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.
Step 1: Calculate the volume of the hemisphere.
Volume of hemisphere = (2/3)πr³
= (2/3)π(1 cm)³
= (2/3)π cm³
Step 2: Calculate the volume of the cone.
Volume of cone = (1/3)πr²h
Since the height (h) is equal to the radius (r), h = r = 1 cm.
= (1/3)π(1 cm)²(1 cm)
= (1/3)π cm³
Step 3: Find the total volume of the solid.
Total volume = Volume of hemisphere + Volume of cone
= (2/3)π cm³ + (1/3)π cm³
= π cm³
The volume of the solid in terms of π is π cm³.
2. 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 to be nearly the same.)
Step 1: Calculate the volume of the cylinder.
Radius of the cylinder (r) = Diameter / 2 = 3 cm / 2 = 1.5 cm
Height of the cylinder (h) = Total length – 2 × height of cones = 12 cm – 2(2 cm) = 8 cm
Volume of cylinder = πr²h
= π(1.5 cm)²(8 cm)
= π(2.25 cm²)(8 cm)
= 18π cm³
Step 2: Calculate the volume of both cones.
Volume of one cone = (1/3)πr²h
= (1/3)π(1.5 cm)²(2 cm)
= (1/3)π(2.25 cm²)(2 cm)
= 1.5π cm³
Volume of two cones = 2 × Volume of one cone
= 2 × 1.5π cm³
= 3π cm³
Step 3: Find the total volume of air contained in the model.
Total volume = Volume of cylinder + Volume of two cones
= 18π cm³ + 3π cm³
= 21π cm³
The volume of air contained in the model is 21π cm³.
3. 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 Fig. 12.15).
Step 1: Calculate the volume of one gulab jamun.
Radius (r) = Diameter / 2 = 2.8 cm / 2 = 1.4 cm
Length of the cylindrical part (h) = Total length – Diameter = 5 cm – 2.8 cm = 2.2 cm
Volume of the cylindrical part = πr²h
= π(1.4 cm)²(2.2 cm)
= π(1.96 cm²)(2.2 cm)
= 4.312π cm³
Volume of two hemispheres = Volume of one sphere
= (4/3)πr³
= (4/3)π(1.4 cm)³
= (4/3)π(2.744 cm³)
= 3.648π cm³
Total volume of one gulab jamun = Volume of cylindrical part + Volume of two hemispheres
= 4.312π cm³ + 3.648π cm³
= 7.96π cm³
Step 2: Calculate the volume of syrup in one gulab jamun.
Volume of syrup = 30% of total volume
= 0.3 × 7.96π cm³
= 2.388π cm³
Step 3: Find the total volume of syrup in 45 gulab jamuns.
Total volume of syrup = 45 × Volume of syrup in one gulab jamun
= 45 × 2.388π cm³
≈ 107.46π cm³
The total volume of syrup in 45 gulab jamuns is approximately 107.46π cm³.
4. 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. 12.16).
Step 1: Calculate the volume of the cuboid.
Volume of cuboid = length × breadth × height
= 15 cm × 10 cm × 3.5 cm
= 525 cm³
Step 2: Calculate the volume of one conical depression.
Volume of cone = (1/3)πr²h
= (1/3)π(0.5 cm)²(1.4 cm)
= (1/3)π(0.25 cm²)(1.4 cm)
= (1/3)π(0.35 cm³)
= (1/3) × (22/7) × 0.35 cm³
= (1/3) × 22/7 × 0.35 cm³
= 0.3667 cm³ (approximately)
Step 3: Calculate the volume of wood removed by the four conical depressions.
Total volume of wood removed = 4 × Volume of one conical depression
= 4 × 0.3667 cm³
= 1.4668 cm³ (approximately)
Step 4: Calculate the volume of wood in the entire pen stand.
Volume of wood = Volume of cuboid – Volume of wood removed by conical depressions
= 525 cm³ – 1.4668 cm³
= 523.5332 cm³ (approximately)
The volume of wood in the entire pen stand is approximately 523.53 cm³.
5. 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.
Step 1: Calculate the volume of the cone.
Volume of cone = (1/3)πr²h
= (1/3)π(5 cm)²(8 cm)
= (1/3)π(25 cm²)(8 cm)
= (1/3) × 3.14 × 200 cm³
= 209.33 cm³ (using π = 3.14)
Step 2: Determine the volume of water that flows out when the lead shots are dropped in.
Volume of water flowed out = (1/4) × Volume of cone
= (1/4) × 209.33 cm³
= 52.3325 cm³
Step 3: Calculate the volume of one lead shot.
Volume of one lead shot = (4/3)πr³
= (4/3)π(0.5 cm)³
= (4/3) × 3.14 × 0.125 cm³
= 0.5233 cm³ (using π = 3.14)
Step 4: Find the number of lead shots dropped in the vessel.
Number of lead shots = Volume of water flowed out / Volume of one lead shot
= 52.3325 cm³ / 0.5233 cm³
≈ 100
100 lead shots are dropped in the vessel.
6. 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 8g mass. (Use π = 3.14)
Step 1: Calculate the volume of the larger cylinder (base cylinder).
Radius of the larger cylinder (r₁) = Diameter / 2 = 24 cm / 2 = 12 cm
Volume of the larger cylinder = πr₁²h₁
= 3.14 × (12 cm)² × 220 cm
= 3.14 × 144 cm² × 220 cm
= 99168 cm³
Step 2: Calculate the volume of the smaller cylinder (top cylinder).
Radius of the smaller cylinder (r₂) = 8 cm
Volume of the smaller cylinder = πr₂²h₂
= 3.14 × (8 cm)² × 60 cm
= 3.14 × 64 cm² × 60 cm
= 12057.6 cm³
Step 3: Find the total volume of the pole.
Total volume = Volume of the larger cylinder + Volume of the smaller cylinder
= 99168 cm³ + 12057.6 cm³
= 111225.6 cm³
Step 4: Calculate the mass of the pole.
Mass of the pole = Volume × Density of iron
= 111225.6 cm³ × 8g/cm³
= 889804.8 g
The mass of the pole is approximately 889.8 kg (since 1 kg = 1000 g).
7. 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.
Step 1: Calculate the volume of the cone.
Volume of cone = (1/3)πr²h
= (1/3)π(60 cm)²(120 cm)
= (1/3) × 3.14 × 3600 cm² × 120 cm
= 452160 cm³
Step 2: Calculate the volume of the hemisphere.
Volume of hemisphere = (2/3)πr³
= (2/3) × 3.14 × (60 cm)³
= (2/3) × 3.14 × 216000 cm³
= 452160 cm³
Step 3: Calculate the volume of the cylinder.
Volume of cylinder = πr²h
= 3.14 × (60 cm)² × 180 cm
= 3.14 × 3600 cm² × 180 cm
= 2030400 cm³
Step 4: Find the volume of water left in the cylinder.
Volume of water left = Volume of cylinder – (Volume of cone + Volume of hemisphere)
= 2030400 cm³ – (452160 cm³ + 452160 cm³)
= 2030400 cm³ – 904320 cm³
= 1126080 cm³
8. 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 cm³. Check whether she is correct, taking the above as the inside measurements, and π = 3.14.
Step 1: Calculate the volume of the cylindrical neck.
Radius of neck (r) = Diameter / 2 = 2 cm / 2 = 1 cm
Volume of cylindrical neck = πr²h
= 3.14 × (1 cm)² × 8 cm
= 25.12 cm³
Step 2: Calculate the volume of the spherical part.
Radius of sphere (R) = Diameter / 2 = 8.5 cm / 2
= 4.25 cm
Volume of sphere = (4/3)πR³
= (4/3) × 3.14 × (4.25 cm)³
= 321.7625 cm³
Step 3: Calculate the total volume of the vessel.
Total volume = Volume of cylindrical neck + Volume of sphere
= 25.12 cm³ + 321.7625 cm³
= 346.8825 cm³
The child’s measurement of 345 cm³ is close to our calculated volume, so her measurement is approximately correct.