An empty cone is of radius 3 cm and height 12 cm. Ice-cream is filled in it so that lower part of the cone which is 1/6th of the volume of the cone is unfilled but hemisphere is formed on the top. Find volume of the ice-cream. (Take π = 3.14)

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

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

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

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

An empty cone is of radius 3 cm and height 12 cm. Ice-cream is filled in it so that lower part of the cone which is $\frac{1}{6}$ th of the volume of the cone is unfilled but hemisphere is formed on the top. Find volume of the ice-cream. (Take $π = 3.14$ )

Figure in the paper: An inverted cone (vertex at bottom) with an ellipse at the top rim, centre marked. A hemisphere sits on top of the cone's open rim. Given: cone radius 3 cm, height 12 cm. See it in the official paper.

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

The ice-cream is the filled part of the cone plus a hemisphere of the same radius.

Hint 2

Filled part = $\frac{5}{6}$ of the cone's volume. Use $\frac{2}{3} π r^{3}$ for the 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.

Volume of cone $= \frac{1}{3} π r^{2} h = \frac{1}{3} \times π \times 3^{2} \times 12 = 36 π$ cm³.
1 mark
$\frac{1}{6}$ of the cone is unfilled, so the ice-cream inside the cone is
$\frac{5}{6} \times 36 π = 30 π$ cm³.
½ mark
Volume of the hemisphere on top (radius 3 cm):
$\frac{2}{3} π (3)^{3} = 18 π$ cm³.
1 mark
Total ice-cream $= 30 π + 18 π = 48 π = 48 \times 3.14 = 150.72$ cm³.
½ mark

Answer: 150.72 cm³

Where the marks are

Volume of cone $= 36 π$ cm³ (1 mark)
Ice-cream in cone $= \frac{5}{6}$ of cone $= 30 π$ cm³ (½ mark)
Hemisphere volume $= 18 π$ cm³ (1 mark)
Total $= 48 π = 150.72$ cm³ (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(b)

Hints

Step-by-step solution

Revise the chapter

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