MathsCBSE Class 10 · Basic · Board exam 2025, set 430/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2025: question 34

Set in the CBSE Class 10 board exam of 2025, 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: A cylindrical bottle of height 10 cm, with a hemispherical portion raised from the base into the interior of the bottle. The height of 10 cm is marked along the outer side of the cylinder. A small nozzle sits on top of the bottle. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 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 away space from the cylinder, so subtract its volume.

Hint 2

The hemisphere has the same radius as the bottle, 2.5 cm; use $\frac{2}{3} π r^{3}$ for it.

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.

Radius of cylinder = radius of hemisphere $= r = \frac{5}{2} = 2.5$ cm; height of cylinder $h = 10$ cm.
Capacity = Volume of cylinder − Volume of hemisphere $= π r^{2} h - \frac{2}{3} π r^{3}$
½ mark
Volume of cylinder $= 3.14 \times \frac{5}{2} \times \frac{5}{2} \times 10 = 196.25 cm^{3}$
1½ marks
Volume of hemisphere $= \frac{2}{3} \times 3.14 \times \frac{5}{2} \times \frac{5}{2} \times \frac{5}{2} = \frac{785}{24} \approx 32.71 cm^{3}$
1½ marks
Capacity $= 196.25 - 32.71 = \frac{3925}{24} \approx 163.5 cm^{3}$
1 mark
Since $1 cm^{3} = 1$ mL, capacity of the bottle ≈ 163.5 mL.
½ mark

Answer: ≈ 163.5 mL (3925/24 cm³)

Where the marks are

Identifies r = 2.5 cm, h = 10 cm and capacity = cylinder − hemisphere (½ mark)
Correct volume of the cylinder, 196.25 cm³ (1½ marks)
Correct volume of the hemisphere, ≈ 32.71 cm³ (1½ marks)
Subtracts to get ≈ 163.5 cm³ (1 mark)
Converts to 163.5 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 paper2025 board paper, set 430/3/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 34

Hints

Step-by-step solution

Revise the chapter

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