Daily-life problems on combined solids
Read the story, name the two solids, then decide: paint, cloth or polish means surface area; holds, fills or wood used means volume.
Pathway: solving a word problem on solids
- 1Read the question twice and name the solids, for example 'cone on a hemisphere'.
- 2Draw a rough figure and write every length on it.
- 3Find missing lengths: radius = diameter ÷ 2, cone height = total height − r, slant height l = √(r² + h²).
- 4Decide: surface area (paint, cloth) or volume (holds, made of).
- 5Write the formula, substitute and simplify.
- 6Multiply by the rate if a cost is asked. Change cm³ to litres if needed.
- 7Write the answer with its unit and box it.
Worked example 1: ice cream cones
An ice cream cone has radius 3.5 cm and height 8 cm. It is filled with ice cream and has a hemisphere of ice cream of the same radius on top. Find the volume of ice cream in one cone. How much ice cream is needed for 40 such cones? (Use π = 22/7.)
- Given: r = 3.5 cm, cone height h = 8 cm.
- Formula: ice cream = cone + hemisphere = (1/3)πr²h + (2/3)πr³ = (1/3)πr²(h + 2r).
- πr² = (22/7) × 3.5 × 3.5 = 38.5 cm².
- h + 2r = 8 + 7 = 15 cm.
- Volume = (1/3) × 38.5 × 15 = 38.5 × 5 = 192.5 cm³.
- For 40 cones: 192.5 × 40 = 7700 cm³ = 7700 ÷ 1000 = 7.7 litres.
One cone has 192.5 cm³ of ice cream; 40 cones need 7.7 litres.
Worked example 2: painting Channapatna tops
A wooden top is a cone on a hemisphere, both of radius 3.5 cm. The slant height of the cone is 5 cm. The cone is painted red and the hemisphere yellow. Find the area of each colour, and the cost of painting 100 tops at ₹ 1 per 10 cm². (Use π = 22/7.)
- Formula: red area = CSA of cone = πrl; yellow area = CSA of hemisphere = 2πr².
- Red area = (22/7) × 3.5 × 5 = 11 × 5 = 55 cm².
- Yellow area = 2 × (22/7) × 3.5 × 3.5 = 2 × 38.5 = 77 cm².
- Area of one top = 55 + 77 = 132 cm².
- Area of 100 tops = 132 × 100 = 13200 cm².
- Cost = (13200 ÷ 10) × ₹ 1 = ₹ 1320.
Red 55 cm² and yellow 77 cm² on each top; painting 100 tops costs ₹ 1320.
Which measure does the question want?
| The question says | Find | Example |
|---|---|---|
| paint, polish, whitewash | Surface area (outside only, unless the inside is also asked) | Painting a dome or a toy |
| canvas, cloth, cover | Surface area (usually no floor) | Canvas for a tent |
| holds, capacity, fill | Volume, often in litres | Water in a tank, milk in a vessel |
| made of, material, wood or metal used | Volume of the solid part | Wood left in a pen stand |
Common traps
- Diameter given, radius needed: always halve it first.
- Total height includes the hemisphere: subtract r to get the cone height.
- Mixed units: change everything to cm, or everything to m, before you start.
- A tent has no floor canvas unless the question says so.
- Rate per 10 cm² or per 100 cm²: divide the area first, then multiply by the rate.
A kulfi seller, a shamiana (tent) maker and a Channapatna toy painter solve these problems every day. Choosing area or volume correctly saves material and money.
3 marks: “A kulfi is a cone of radius 3 cm and height 6 cm with a hemisphere of the same radius on top. Find the volume of the kulfi. (Use π = 3.14.)” V = (1/3)πr²(h + 2r) = (1/3) × 3.14 × 9 × 12 = 3.14 × 36 = 113.04 cm³.
5 marks: “A water tank is a cylinder of radius 2.1 m and height 4 m, closed at the top, with a hemispherical bottom of the same radius. (i) How many litres does it hold? (ii) Find the cost of painting its outer surface at ₹ 50 per m². (Use π = 22/7.)” (i) V = πr²h + (2/3)πr³ = 13.86 × (4 + 1.4) = 74.844 m³ = 74,844 litres. (ii) Area = πr² + 2πrh + 2πr² = 13.86 + 52.8 + 27.72 = 94.38 m²; cost = 94.38 × 50 = ₹ 4719.
Paint and cloth go on the skin (area). Water and wood fill the inside (volume).