MathsCBSE Class 10 · Basic · Board exam 2023, set 430/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 35

Set in the CBSE Class 10 board exam of 2023, paper 430/4/1, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerSurface Areas and Volumes

The boilers are used in thermal power plants to store water and then used to produce steam. One such boiler consists of a cylindrical part in middle and two hemispherical parts at its both ends. Length of the cylindrical part is $7$ m and radius of cylindrical part is $\frac{7}{2}$ m. Find the total surface area and the volume of the boiler. Also, find the ratio of the volume of cylindrical part to the volume of one hemispherical part.

Figure in the paper: A boiler shaped as a cylinder standing upright with a hemisphere attached to its top and a hemisphere attached to its bottom. The cylindrical middle part has height 7 m, and its radius is marked as $\frac{7}{2}$ m from the centre of the top circular face to its rim. The height of the cylindrical part is labelled 7 m. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/4/1.

Hints

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

Hint 1

The outer surface is the curved surface of the cylinder plus the curved surfaces of two hemispheres. The flat circular faces are hidden.

Hint 2

Add the volume of the cylinder, $π r^{2} h$ , to the volume of two hemispheres, $2 \times \frac{2}{3} π r^{3}$ .

Hint 3

For the ratio, cancel $π r^{2}$ before you substitute the values.

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.

Here $h = 7$ m and $r = \frac{7}{2}$ m.
TSA of boiler = CSA of cylinder + CSA of 2 hemispheres $= 2 π r h + 2 (2 π r^{2}) = 2 π r (h + 2 r)$
$= 2 \times \frac{22}{7} \times \frac{7}{2} (7 + 2 \times \frac{7}{2}) = 22 \times 14$
1½ marks
Total surface area $= 308 m^{2}$
½ mark
Volume of boiler $= π r^{2} h + 2 (\frac{2}{3} π r^{3})$
$= \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 7 + \frac{4}{3} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}$
$= 269.5 + \frac{1078}{6}$
1½ marks
$= \frac{1617}{6} + \frac{1078}{6} = \frac{2695}{6} m^{3} \approx 449.17 m^{3}$
½ mark
$\frac{Volume of cylindrical part}{Volume of one hemisphere} = \frac{π r ^{2} h}{\frac{2}{3} π r ^{3}}$
½ mark
$= \frac{3 h}{2 r} = \frac{3 \times 7}{2 \times \frac{7}{2}} = \frac{21}{7} = \frac{3}{1}$
So the required ratio is $3 : 1$ .
½ mark

Answer: TSA = 308 m²; Volume = 2695/6 m³ ≈ 449.17 m³; ratio 3 : 1

Where the marks are

Correct TSA formula with substitution (1½ marks)
TSA = 308 m² (½ mark)
Correct volume formula with substitution (1½ marks)
Volume = 2695/6 m³ ≈ 449.17 m³ (½ mark)
Sets up the ratio $π r^{2} h : \frac{2}{3} π r^{3}$ (½ mark)
Ratio = 3 : 1 (½ 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 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 35

Hints

Step-by-step solution

Revise the chapter

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