MathsCBSE Class 10 · Basic · Board exam 2023, set 430/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 33

Set in the CBSE Class 10 board exam of 2023, paper 430/1/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerCompetency-basedSurface Areas and Volumes

A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. Also, find the volume of the vessel.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/1/1.

Hints

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

Hint 1

The total height includes the hemisphere's radius; subtract it to get the cylinder's height.

Hint 2

Inner surface = curved surface of cylinder + curved surface of hemisphere; volume = cylinder + hemisphere.

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 hemisphere = radius of cylinder $r = \frac{14}{2} = 7$ cm. Height of cylinder $h = 13 - 7 = 6$ cm.
1 mark
Inner surface area = CSA of cylinder + CSA of hemisphere $= 2 π r h + 2 π r^{2} = 2 π r (h + r) = 2 \times \frac{22}{7} \times 7 \times (6 + 7)$
1 mark
$= 44 \times 13 = 572 cm^{2}$
1 mark
Volume = volume of cylinder + volume of hemisphere $= π r^{2} h + \frac{2}{3} π r^{3} = π r^{2} (h + \frac{2}{3} r) = \frac{22}{7} \times 7 \times 7 \times (6 + \frac{14}{3})$
1 mark
$= 154 \times \frac{32}{3} = \frac{4928}{3} cm^{3} \approx 1642.67 cm^{3}$
1 mark

Answer: Inner surface area = 572 cm²; Volume = 4928/3 cm³ ≈ 1642.67 cm³

Where the marks are

Finds $r = 7$ cm and cylinder height 6 cm (1 mark)
Uses $2 π r h + 2 π r^{2}$ with substitution (1 mark)
Inner surface area $= 572$ cm² (1 mark)
Uses $π r^{2} h + \frac{2}{3} π r^{3}$ with substitution (1 mark)
Volume $= \frac{4928}{3}$ cm³ (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Surface Area 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 430/1/1 (Basic): 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 Basic 2023: question 33

Hints

Step-by-step solution

Revise the chapter

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