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

CBSE Class 10 Mathematics Standard 2024: question 31

Set in the CBSE Class 10 board exam of 2024, paper 30/1/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerSurface Areas and Volumes

The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm. If the volume of the metal used in making the cylinder is 176 cm $^{3}$ , find the outer and inner radii of the cylinder.

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

Hints

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

Hint 1

The volume of metal is the outer cylinder's volume minus the inner cylinder's volume.

Hint 2

Factorise $R^{2} - r^{2}$ and use the given difference of the radii.

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 the outer radius be $R$ cm and the inner radius be $r$ cm. Then $R - r = 1$ ...(1)
½ mark
Volume of metal $= π h (R^{2} - r^{2})$ :
$\frac{22}{7} \times 14 \times (R^{2} - r^{2}) = 176 \Rightarrow 44 (R^{2} - r^{2}) = 176 \Rightarrow R^{2} - r^{2} = 4$ .
1 mark
$(R + r) (R - r) = 4$ . Using (1), $R + r = 4$ ...(2)
½ mark
Add (1) and (2): $2 R = 5 \Rightarrow R = 2.5$ cm. Then $r = 1.5$ cm.
Outer radius = 2.5 cm, inner radius = 1.5 cm.
1 mark

Answer: Outer radius 2.5 cm, inner radius 1.5 cm

Where the marks are

Writes $R - r = 1$ (½ mark)
Sets up $\frac{22}{7} \times 14 \times (R^{2} - r^{2}) = 176$ (1 mark)
Obtains $R + r = 4$ (½ mark)
Solves to get $R = 2.5$ cm and $r = 1.5$ cm (1 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 paper2024 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 31

Hints

Step-by-step solution

Revise the chapter

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