MathsCBSE Class 10 · Standard · Board exam 2024, set 30/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 35(b)

Set in the CBSE Class 10 board exam of 2024, paper 30/3/1, section D, question 35(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(b) · 5 marks · Long answerCompetency-basedSurface Areas and Volumes

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 4 mm, find its surface area. Also, find its volume.

Figure in the paper: A capsule shape: a cylinder with a hemisphere at each end. Overall length marked 14 mm (horizontal) and diameter marked 4 mm (vertical) at the left end. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/3/1.

Hints

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

Hint 1

Split the capsule into one cylinder and two hemispheres.

Hint 2

The length of the cylinder part is the total length minus the two radii.

Hint 3

Surface area uses $2 π r h + 2 (2 π r^{2})$ . Volume uses $π r^{2} h + 2 \cdot \frac{2}{3} π r^{3}$ .

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.

Radius of each hemisphere $=$ radius of the cylinder $= \frac{4}{2} = 2$ mm
½ mark
Length of the cylindrical part $= 14 - 2 - 2 = 10$ mm
½ mark
Surface area $=$ CSA of cylinder $+ 2 \times$ CSA of hemisphere $= 2 π r h + 2 (2 π r^{2})$
$= 2 \times \frac{22}{7} \times 2 \times 10 + 2 \times 2 \times \frac{22}{7} \times 2 \times 2 = \frac{880}{7} + \frac{352}{7}$
1 mark
$= \frac{1232}{7} = 176 mm^{2}$
1 mark
Volume $=$ volume of cylinder $+ 2 \times$ volume of hemisphere $= π r^{2} h + 2 \times \frac{2}{3} π r^{3}$
$= \frac{22}{7} \times 2 \times 2 \times 10 + 2 \times \frac{2}{3} \times \frac{22}{7} \times 2 \times 2 \times 2 = \frac{880}{7} + \frac{704}{21}$
1 mark
$= \frac{2640 + 704}{21} = \frac{3344}{21} \approx 159.24 mm^{3}$
1 mark

Answer: Surface area $= 176$ mm²; Volume $= \frac{3344}{21}$ mm³ $\approx 159.24$ mm³

Where the marks are

Radius 2 mm (½ mark)
Cylinder length 10 mm (½ mark)
Correct surface-area expression (1 mark)
Surface area 176 mm² (1 mark)
Correct volume expression (1 mark)
Volume 3344/21 ≈ 159.24 mm³ (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Surface Area and Volume Formulae Revision

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 paper2024 board paper, set 30/3/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 35(b)

Hints

Step-by-step solution

Revise the chapter

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