MathsCBSE Class 10 · Standard · Board exam 2025, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 30

Set in the CBSE Class 10 board exam of 2025, paper 30/1/1, section C, question 30, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q30 · 3 marks · Short answerCompetency-basedSurface Areas and Volumes

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains $\frac{1408}{21} m^{3}$ of air, find the height of the cylindrical part. (Use $π = \frac{22}{7}$ ).

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/1/1.

Hints

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

Hint 1

The air in the room fills a cylinder and a hemisphere with the same radius.

Hint 2

Write the height $h$ in terms of $r$ , then add the two volume formulas.

Hint 3

Equate the total to $\frac{1408}{21}$ and solve for $r^{3}$ .

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.

Let $r$ be the radius of the hemisphere (which is also the radius of the cylinder) and $h$ the height of the cylinder. Given $r = \frac{h}{2}$ , so $h = 2 r$ .
½ mark
Volume of air = volume of cylinder + volume of hemisphere $= π r^{2} h + \frac{2}{3} π r^{3}$ . So $\frac{1408}{21} = π r^{2} (2 r) + \frac{2}{3} π r^{3}$ .
½ mark
Simplify: $2 π r^{3} + \frac{2}{3} π r^{3} = \frac{8}{3} π r^{3}$ . So $\frac{8}{3} \times \frac{22}{7} \times r^{3} = \frac{1408}{21}$ , which gives $\frac{176}{21} r^{3} = \frac{1408}{21}$ . So $r^{3} = 8$ .
1 mark
Therefore $r = 2$ m.
½ mark
Height of the cylindrical part $h = 2 r = 4$ m.
½ mark

Answer: Height of the cylindrical part = 4 m

Where the marks are

Writes $h = 2 r$ (½ mark)
Writes the total volume as cylinder + hemisphere and equates it to $\frac{1408}{21}$ (½ mark)
Simplifies to get $r^{3} = 8$ (1 mark)
Gets $r = 2$ m (½ mark)
Gets $h = 4$ m (½ mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Volume of a Combination of Solids

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 30/1/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2025: question 30

Hints

Step-by-step solution

Revise the chapter

More questions from Surface Areas and Volumes