Find the volume of ink in a new ball point pen. Take the necessary measurements and make approximations, as needed.
Answer: The ink sits in the refill tube, a thin cylinder. With inside diameter about 2 mm (r ≈ 0.1 cm) and an ink column about 10 cm long: V ≈ 3.14 × 0.12 × 10 ≈ 0.3 cm3 (about 0.3 mL).
Step-by-step solution
Idea: Model the ink as a long thin cylinder inside the refill. Measure the length of the ink column and estimate the inner diameter of the tube.
- Measure a new refill: ink column ≈ 10 cm long. The inside of the tube is about 2 mm across, so r ≈ 1 mm = 0.1 cm (the tip holds a negligible amount).1 mark
- Volume ≈ πr2h = 3.14 × 0.1 × 0.1 × 10 = 0.314 cm3.1 mark
- So a new pen holds about 0.3 cm3 = 0.3 mL of ink, less than one-tenth of a teaspoon.1 mark
About 0.3 cm³ (0.3 mL) of ink, taking the ink as a cylinder 10 cm long with inner radius 0.1 cm.
Check: A 1 mm error in the diameter matters a lot (radius is squared): with r = 0.12 cm, V ≈ 0.45 cm3. Either way it is a fraction of 1 mL ✓.
Answer to write in the exam
Ink column: h ≈ 10 cm; inner radius r ≈ 0.1 cm
V = πr2h ≈ 3.14 × 0.01 × 10
∴ V ≈ 0.31 cm3 ≈ 0.3 mL
Common mistakes that cost marks
- Using the outside diameter of the pen body (about 1 cm), which gives about 8 cm3, far too much.
- Using the diameter as the radius.
- Forgetting to convert mm to cm before squaring.
How this can come in the exam
Short answer (2 marks)
A refill has ink in a cylinder of inner diameter 1.4 mm and length 12 cm. Find the volume of ink in mm3. (π = 227)
Show answer
r = 0.7 mm, h = 120 mm: V = 227 × 0.49 × 120 = 184.8 mm3 (≈ 0.18 mL).Try one yourself
A fountain pen’s ink tube is a cylinder 6 cm long with inner radius 0.25 cm. How much ink does it hold? (π = 3.14)
Show answer
3.14 × 0.0625 × 6 ≈ 1.18 cm3 (about 1.2 mL).
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