MathsCBSE Class 10 · Basic · Board exam 2026, set 430/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2026: question 34

Set in the CBSE Class 10 board exam of 2026, paper 430/3/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerCompetency-basedSurface Areas and Volumes

A perfume bottle is in the form of a cylinder but the bottom of the bottle has a hemispherical raised portion to reduce the capacity of the bottle. The inner diameter of the bottle is 5 cm and the height of the bottle is 10 cm. Find the capacity of the perfume bottle in mL. (Use $π = 3.14$ and $1 c m^{3} = 1 mL$ )

Figure in the paper: Cylindrical bottle of height 10 cm with a hemispherical portion projecting up from the base into the inside of the bottle; the bottle's top has a small neck/cap. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/3/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The raised hemisphere takes up space inside the cylinder.

Hint 2

The hemisphere's radius equals the bottle's inner radius.

Hint 3

Capacity = cylinder volume − hemisphere volume. Then convert cm³ to mL.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Inner radius $r = \frac{5}{2} = 2.5$ cm and height $h = 10$ cm. The hemispherical raised portion has the same radius, 2.5 cm.
½ mark
Volume of the cylinder $= π r^{2} h = 3.14 \times 2.5 \times 2.5 \times 10 = 196.25 cm^{3}$ .
1½ marks
Volume of the hemisphere $= \frac{2}{3} π r^{3} = \frac{2}{3} \times 3.14 \times 15.625 \approx 32.71 cm^{3}$ .
1½ marks
Capacity $=$ volume of cylinder $-$ volume of hemisphere $= 196.25 - 32.71 = 163.54 cm^{3}$ .
1 mark
Since $1 cm^{3} = 1$ mL, the capacity of the bottle is approximately $163.54$ mL.
½ mark

Answer: Capacity ≈ 163.54 mL

Where the marks are

Takes radius 2.5 cm for both the cylinder and the hemisphere (½ mark)
Volume of cylinder = 196.25 cm³ (1½ marks)
Volume of hemisphere ≈ 32.71 cm³ (1½ marks)
Subtracts to get ≈ 163.54 cm³ (1 mark)
Converts to ≈ 163.54 mL (½ mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Surface Areas and Volumes, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 430/3/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2026: question 34

Hints

Step-by-step solution

Revise the chapter

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