Introduction
In everyday life, we constantly interact with various solid objects—boxes, cylinders, spheres, cones, and more. Chapter 19 delves into practical word problems involving these solids, helping you connect geometry with real‑world applications. In this overview, you will find a clear breakdown of the key concepts, problem types, and step‑by‑step examples using relevant keywords like volume, surface area, solid objects, prism, cylinder, sphere, composite solids, and real‑life applications.
1. Understanding Solids and Their Measures
Begin by recalling key formulae:
- Volume of a prism (rectangular, triangular, etc.): Base area × height
- Cylinder: V = πr²h, Surface area (total) = 2πr(h + r)
- Sphere: V = (4/3)πr³, Surface area = 4πr²
- Cone: V = (1/3)πr²h, Lateral surface area = πrl, Total surface area = πr(l + r)
These formulae are foundational in solving real‑life problems related to solid objects, especially in composite solids where two or more shapes are combined. Understanding volume, surface area, and their real‑life significance—for example, determining capacity, packaging, or material requirements—is essential.
2. Word Problems on Individual Solids
2.1 Cylinders
Example: A water tank is shaped like a cylinder with radius 3 m and height 5 m.
- Find the volume: V = π × 3² × 5 = 45π m³.
- Use in real life: Helps estimate how much water the tank can hold.
Example: A cylindrical gift box has radius 7 cm and height 10 cm.
- Find total surface area: TSA = 2πr(h + r) = 2π × 7 × (10 + 7) = 238π cm².
- Use in real life: Determines how much wrapping paper is needed.
These problems reinforce keywords such as cylinder volume, surface area of cylinder, and real‑life cylinder problems.
2.2 Cones and Spheres
Cone example: A traffic cone has a radius of 0.25 m and slant height of 0.5 m.
- Find lateral surface area: LSA = πr l = π × 0.25 × 0.5 = 0.125π m².
- Use: Useful for calculating reflective tape needed around cones.
Sphere example: A metal ball has radius 4 cm.
- Volume: V = (4/3)π × 4³ = (4/3)π × 64 ≈ 85.3π cm³.
- Use: Helps determine the amount of material used in manufacturing balls.
These problems integrate cone surface area, sphere volume, solid geometry in real life, and manufacturing.
3. Composite Solid Objects
Real‑life shapes are often combinations: cake packages, water filters, toy blocks, and more. You apply additive or subtractive approaches:
Example 1: A solid made by stacking a cylinder (radius 2 cm, height 5 cm) on top of a hemisphere (same radius).
- Cylinder volume: V₁ = π × 2² × 5 = 20π cm³
- Hemisphere volume: V₂ = (1/2) × (4/3)π × 2³ = (1/2) × (32/3)π ≈ (16/3)π cm³
- Total volume: V_total = 20π + (16/3)π = (76/3)π cm³
- Use: Ideal for figuring out capacity of combined shapes like decorative items or container lids.
Example 2: A solid block from which a cylinder (radius 1 m, height 2 m) has been drilled out of a cuboid (2 m × 1 m × 1 m).
- Cuboid volume: 2 × 1 × 1 = 2 m³
- Cylinder volume removed: π × 1² × 2 = 2π m³
- Remaining volume: 2 – 2π m³ (noting π ≈ 3.1416, so you may find a negative value, prompting reflection on realistic physical situations or constraints).
Composite examples support keywords like composite solid volume, combined shape problem, volume subtraction, and engineering applications.
4. Real‑Life Applications
This chapter enhances problem‑solving skills tied to everyday contexts:
- Packaging: Determining the size of material needed for cylindrical containers or cone‑topped jars.
- Construction: Calculating volumes of concrete in columns (cylinders) or spherical domes.
- Manufacturing: Estimating material for metal balls, conical tips, or combined solid parts.
- Domestic use: Figuring out how much water (cylinder tank) or paint (surface area) is required.
Such applications reinforce the connection between theoretical geometry and practical problem solving, bridging the gap between academic syllabus and real-world maths.
5. Tips for Solving Real‑Life Problems with Solids
- Identify shapes: Break complex objects into known solids (cylinder, cone, sphere).
- Write formulae clearly for each part.
- Convert units where needed (e.g., m to cm).
- Add or subtract volumes logically for composite solids.
- Verify your result’s reasonableness—e.g., ensuring volume remains positive.
- Interpret answer in context: “The cistern can hold approximately ___ litres.”
These strategies help build methodical and confident problem‑solving, staying well within the West Bengal board syllabus.
Conclusion
Chapter 19: Real‑Life Problems Related to Different Solid Objects offers an insightful bridge between geometry and its everyday utilities—tanks, packaging, domes, toys, drains, and more. With a clear understanding of volume, surface area, and composite solids, students gain both mathematical precision and real‑world awareness. The structured examples and strategy pointers fosters accuracy, comprehension, and exam readiness—entirely in line with the Class 10 (WB Board) syllabus.