Case Study 2. In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter 7 cm and height 14 cm and the other is hemispherical with diameter 21 cm. (i) Find the area of the base of the cylindrical cup. (ii) (a) What is the capacity of the hemispherical cup ? (ii) (b) Find the capacity of the cylindrical cup. (iii) What is the curved surface area of the cylindrical cup ?

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/2/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2023: question 37

Set in the CBSE Class 10 board exam of 2023, paper 30/2/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedSurface Areas and Volumes

Case Study 2. In a coffee shop, coffee is served in two types of cups. One is cylindrical in shape with diameter $7$ cm and height $14$ cm and the other is hemispherical with diameter $21$ cm.

(i)
Find the area of the base of the cylindrical cup.
1 mark
(ii)
(a) What is the capacity of the hemispherical cup ?
2 marks
OR
(ii)
(b) Find the capacity of the cylindrical cup.
2 marks
(iii)
What is the curved surface area of the cylindrical cup ?
1 mark

Figure in the paper: Two photographs of coffee cups: on the left a cylindrical mug with a handle on its right side; on the right a hemispherical bowl-shaped cup with a handle on its right side. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/2/1.

Hints

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

Hint 1

Diameters are given in the question, so halve each one to get the radius first.

Hint 2

The base of a cylinder is a circle; its capacity is base area × height.

Hint 3

For the hemisphere, use $\frac{2}{3} π r^{3}$ . For the curved surface of the cylinder, use $2 π r h$ .

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.

(i) Radius of the cylindrical cup $r = \frac{7}{2}$ cm.
Area of base $= π r^{2} = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} = \frac{77}{2} = 38.5$ cm $^{2}$ .
1 mark
(ii)(a) Radius of the hemispherical cup $R = \frac{21}{2}$ cm.
Capacity $= \frac{2}{3} π R^{3} = \frac{2}{3} \times \frac{22}{7} \times \frac{21}{2} \times \frac{21}{2} \times \frac{21}{2}$ .
1 mark
$= \frac{4851}{2} = 2425.5$ cm $^{3}$ .
1 mark
(ii)(b) Capacity of the cylindrical cup $= π r^{2} h = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 14$ .
1 mark
$= 539$ cm $^{3}$ .
1 mark
(iii) Curved surface area of the cylinder $= 2 π r h = 2 \times \frac{22}{7} \times \frac{7}{2} \times 14 = 308$ cm $^{2}$ .
1 mark

Answer: (i) 38.5 cm²; (ii)(a) 2425.5 cm³; (ii)(b) 539 cm³; (iii) 308 cm²

Where the marks are

(i) Base area $π r^{2}$ with r = 3.5 cm gives 38.5 cm² (1 mark)
(ii)(a) Formula $\frac{2}{3} π r^{3}$ with r = 10.5 cm (1 mark); 2425.5 cm³ (1 mark)
(ii)(b) Formula $π r^{2} h$ with r = 3.5 cm, h = 14 cm (1 mark); 539 cm³ (1 mark)
(iii) $2 π r h$ gives 308 cm² (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 paper2023 board paper, set 30/2/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 37

Hints

Step-by-step solution

Revise the chapter

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