Estimation (guesstimates) · 4 marks

Lallan’s golgappa (a small hollow ball made of wheat) stall has a cylindrical vessel with pani (spiced water) in it. The vessel is of radius 10 cm and it has pani up to a height of 50 cm. Estimate the number of golgappas he can serve assuming all golgappas are spheres. Before calculating, make a guess.
Can you guess what number this could be — is it in the hundreds, thousands, or more?

Answer: Pani = π × 10² × 50 = 5000π ≈ 15 708 cm³. With a golgappa of radius 1.5 cm filled 60% (≈ 8.5 cm³ of pani), about 1850 golgappas; with radius 1.75 cm filled 80% (≈ 18 cm³), about 870. So the answer is in the high hundreds to about two thousand, depending on the assumptions.

Step-by-step solution

Given: Vessel: cylinder, radius 10 cm, pani height 50 cm; Golgappas taken as spheres
To find: Estimated number of golgappas served

Idea: Number served = total pani ÷ pani in one golgappa. The pani in one golgappa depends on its size and how full it is, which we must assume.

Guess first: the vessel holds about 15 litres and a golgappa takes about a tablespoon, so perhaps around a thousand: in the thousands rather than the hundreds.½ mark
Total pani = πr²h = π × 10 × 10 × 50 = 5000π ≈ 15 708 cm³.1 mark
Student 1: golgappa radius 1.5 cm: volume = 43π(1.5)³ ≈ 14.14 cm³ (thickness ignored). Filled 60%: 14.14 × 0.6 ≈ 8.5 cm³. Number = 5000π8.5 ≈ 1848, about 1850.1 mark
Student 2: radius 1.75 cm: volume = 43π(1.75)³ ≈ 22.4 cm³. Filled 80%: ≈ 18 cm³. Number = 5000π18 ≈ 873, about 870.1 mark
The two estimates differ because the assumptions (size, how full) differ. The final answer depends on the assumptions, estimations and approximations made.½ mark

About 1850 golgappas (radius 1.5 cm, 60% full) or about 870 (radius 1.75 cm, 80% full); the estimate depends on the assumptions.

Check: 15 708 ÷ 8.5 ≈ 1848 ✓ and 15 708 ÷ 18 ≈ 873 ✓. A 15-litre vessel and a teaspoon-to-tablespoon of pani each makes ‘about a thousand’ sensible ✓.

Answer to write in the exam

Pani in vessel = π × 10² × 50 = 5000π ≈ 15 708 cm³

Student 1: V = 43π(1.5)³ ≈ 14.14 cm³; pani = 0.6 × 14.14 ≈ 8.5 cm³

Number ≈ 5000π8.5 ≈ 1848 ≈ 1850

Student 2: V = 43π(1.75)³ ≈ 22.4 cm³; pani = 0.8 × 22.4 ≈ 18 cm³

Number ≈ 5000π18 ≈ 873 ≈ 870

∴ About 870 to 1850 golgappas, depending on the assumptions

Common mistakes that cost marks

Dividing by the whole golgappa volume, as if each were 100% full of pani.
Using the diameter 20 cm as the radius of the vessel (4 times too much pani).
Thinking there is one ‘right’ answer. A guesstimate is judged by its reasoning and assumptions.

How this can come in the exam

Case-based (4 marks)

A lemonade stall has a cylindrical jar of radius 14 cm filled to 30 cm. Each paper cup is a cylinder of radius 3.5 cm and height 8 cm, filled to 34 of its height.
(i) Find the lemonade in the jar. (ii) Find the lemonade in one cup. (iii) How many cups can be served? (iv) If cups are filled to the brim instead, how many full cups can be served? (π = 227)

Show answer

(i) 227 × 196 × 30 = 18 480 cm³ (1 mark). (ii) 227 × 12.25 × 6 = 231 cm³ (1 mark). (iii) 18 480 ÷ 231 = 80 cups (1 mark). (iv) One full cup = 308 cm³; 18 480 ÷ 308 = 60 cups (1 mark).

Try one yourself

A juice can holds 5 litres. Estimate how many glasses of juice it fills if a glass is a cylinder of radius 3 cm and height 10 cm, filled 80%. (π = 3.14)