MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 31(a)

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section C, question 31(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(a) · 3 marks · Short answerCompetency-basedSurface Areas and Volumes

A room is in the form of cylinder surmounted by a hemi-spherical dome. The base radius of hemisphere is one-half the height of cylindrical part. Find total height of the room if it contains $\frac{1408}{21} m^{3}$ of air. (Take $π = \frac{22}{7}$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

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

Hint 1

Write the cylinder's height in terms of the radius $r$ .

Hint 2

Total volume = $π r^{2} h + \frac{2}{3} π r^{3}$ . The total height is the cylinder's height plus the dome's radius.

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 (and of the cylinder), and $h$ the height of the cylindrical part.
Given $r = \frac{h}{2}$ , so $h = 2 r$ .
½ mark
Volume of room = volume of cylinder + volume of hemisphere:
$π r^{2} (2 r) + \frac{2}{3} π r^{3} = 2 π r^{3} + \frac{2}{3} π r^{3} = \frac{8}{3} π r^{3}$
So $\frac{8}{3} π r^{3} = \frac{1408}{21}$ .
1 mark
$\frac{8}{3} \times \frac{22}{7} \times r^{3} = \frac{1408}{21} \Rightarrow \frac{176}{21} r^{3} = \frac{1408}{21} \Rightarrow r^{3} = 8 \Rightarrow r = 2$ m.
½ mark
Height of the cylindrical part: $h = 2 r = 4$ m.
½ mark
Total height of room $= h + r = 4 + 2 = 6$ m.
½ mark

Answer: 6 m

Where the marks are

Sets up $h = 2 r$ (½ mark)
Writes the total volume $2 π r^{3} + \frac{2}{3} π r^{3} = \frac{1408}{21}$ (1 mark)
Gets $r = 2$ m (½ mark)
Gets $h = 4$ m (½ mark)
Total height = 6 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 paper2023 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 31(a)

Hints

Step-by-step solution

Revise the chapter

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