Case Study 3. A wooden toy is shown in the picture. This is a cuboidal wooden block of dimensions 14\ cm × 17\ cm × 4\ cm. On its top there are seven cylindrical hollows for bees to fit in. Each cylindrical hollow is of height 3 cm and radius 2 cm. Based on the above, answer the following questions: (i) Find the volume of wood carved out to make one cylindrical hollow. (ii) Find the lateral surface area of the cuboid to paint it with green colour. (iii) (a) Find the volume of wood in the remaining cuboid after carving out seven cylindrical hollows. (iii) (b) Find the surface area of the top surface of the cuboid to be painted yellow.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/2/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2023: question 38

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedSurface Areas and Volumes

Case Study 3. A wooden toy is shown in the picture. This is a cuboidal wooden block of dimensions $14 cm \times 17 cm \times 4 cm$ . On its top there are seven cylindrical hollows for bees to fit in. Each cylindrical hollow is of height 3 cm and radius 2 cm.

Based on the above, answer the following questions:

(i)
Find the volume of wood carved out to make one cylindrical hollow.
1 mark
(ii)
Find the lateral surface area of the cuboid to paint it with green colour.
1 mark
(iii)
(a) Find the volume of wood in the remaining cuboid after carving out seven cylindrical hollows.
2 marks
OR
(iii)
(b) Find the surface area of the top surface of the cuboid to be painted yellow.
2 marks

Figure in the paper: Photograph of a yellow cuboidal wooden toy with a honeycomb pattern on its front face. Seven circular hollows are arranged in a honeycomb pattern on the top face, each hollow is coloured-outlined, and colourful bee-shaped rods/pegs stand below the block. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/2/1.

Hints

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

Hint 1

Each hollow is a cylinder: use $π r^{2} h$ .

Hint 2

Lateral surface area of a cuboid is $2 (l + b) h$ , taking the 4 cm side as height.

Hint 3

For the top surface, remove the areas of the seven circular openings from the rectangle's area.

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) Volume of one hollow $= π r^{2} h = \frac{22}{7} \times 2 \times 2 \times 3 = \frac{264}{7}$ cm³ $\approx 37.71$ cm³.
1 mark
(ii) Lateral surface area of cuboid $= 2 (l + b) h = 2 (14 \times 4 + 17 \times 4) = 2 (56 + 68) = 248$ cm².
1 mark
(iii)(a) Volume of 7 hollows $= 7 \times \frac{264}{7} = 264$ cm³.
½ mark
Volume of the full cuboid $= l \times b \times h = 14 \times 17 \times 4 = 952$ cm³.
1 mark
Remaining wood $= 952 - 264 = 688$ cm³.
½ mark
(iii)(b) Top area to be painted = area of top face − area of 7 circular openings $= (14 \times 17) - 7 \times \frac{22}{7} \times 2^{2} = 238 - 88$
1 mark
$= 150$ cm².
1 mark

Answer: (i) 264/7 cm³ ≈ 37.71 cm³; (ii) 248 cm²; (iii)(a) 688 cm³; (iii)(b) 150 cm²

Where the marks are

Volume of one cylindrical hollow = 264/7 cm³ (1 mark)
LSA = 2(14×4 + 17×4) = 248 cm² (1 mark)
(iii)(a) Volume of 7 hollows 264 cm³ (½ mark), cuboid volume 952 cm³ (1 mark), remaining 688 cm³ (½ mark)
(iii)(b) Expression 14×17 − 7πr² with values (1 mark), answer 150 cm² (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Cube, Cuboid and Cylinder Combination Problems

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 430/2/1 (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 38

Hints

Step-by-step solution

Revise the chapter

More questions from Surface Areas and Volumes