MathsCBSE Class 10 · Basic · Board exam 2024, set 430/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 35(a)

Set in the CBSE Class 10 board exam of 2024, paper 430/2/1, section D, question 35(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(a) · 5 marks · Long answerSurface Areas and Volumes

A textile industry runs in a shed. This shed is in the shape of a cuboid surmounted by a half cylinder. If the base of the industry is of dimensions 14 m × 20 m and the height of the cuboidal portion is 7 m, find the volume of air that the industry can hold. Further, suppose the machinery in the industry occupies a total space of 400 m $^{3}$ . Then, how much space is left in the industry?

Figure in the paper: A 3D shape: a cuboid with a semicircular (half-cylinder) roof on top. Front face width is 14 m (the semicircle spans this width, radius 7 m) and height of the cuboidal part is 7 m. The length along the shed is 20 m, marked along the top edge. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/2/1.

Hints

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

Hint 1

Split the shed into two solids: a cuboid at the bottom and half of a cylinder lying on its side on top.

Hint 2

The curved roof spans the 14 m width, so 14 m is its diameter; the 20 m length of the shed is the length of the cylinder.

Hint 3

Add the two volumes to get the air the shed holds, then subtract the space taken by the machinery.

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.

Cuboidal part: length $= 20$ m, breadth $= 14$ m, height $= 7$ m.
Volume of cuboid $= l \times b \times h = 20 \times 14 \times 7 = 1960 m^{3}$
1 mark
Half-cylinder roof: its diameter is the width $14$ m, so $r = \frac{14}{2} = 7$ m. Its length (height of the cylinder) is $h = 20$ m.
1 mark
Volume of half cylinder $= \frac{1}{2} π r^{2} h = \frac{1}{2} \times \frac{22}{7} \times 7 \times 7 \times 20 = 1540 m^{3}$
2 marks
Volume of air the shed can hold $= 1960 + 1540 = 3500 m^{3}$
½ mark
The machinery occupies $400 m^{3}$ .
Space left $= 3500 - 400 = 3100 m^{3}$
½ mark

Answer: Volume of air = 3500 m³; space left = 3100 m³

Where the marks are

Volume of cuboid $= 1960 m^{3}$ (1 mark)
Identifies $r = 7$ m and $h = 20$ m for the half cylinder (1 mark)
Volume of half cylinder $= \frac{1}{2} π r^{2} h = 1540 m^{3}$ (2 marks)
Total volume of air $= 3500 m^{3}$ (½ mark)
Space left $= 3100 m^{3}$ (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 paper2024 board paper, set 430/2/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2024: question 35(a)

Hints

Step-by-step solution

Revise the chapter

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