Playing in a ball pool is good entertainment for kids. Suhana bought 600 new balls of diameter 7 cm to fill in the pool for her kids. The cuboidal box containing 600 balls has dimensions 42 cm × 91 cm × 50 cm (l × b × h). Based on above information, answer the following questions. (i) Find the volume of one ball. (ii) 10 balls are painted with neon colours. Determine the area of painted surface. (iii)(a) Find the volume of empty space in the box. (iii)(b) The lowermost layer of the balls covers the base of the box edge to edge when balls are placed evenly adjacent to each other. (A) How much area is covered by one ball? (B) How many balls are there in lowermost layer?

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 37

Set in the CBSE Class 10 board exam of 2025, paper 430/4/1, 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

Playing in a ball pool is good entertainment for kids. Suhana bought 600 new balls of diameter 7 cm to fill in the pool for her kids. The cuboidal box containing 600 balls has dimensions 42 cm × 91 cm × 50 cm ( $l \times b \times h$ ). Based on above information, answer the following questions.

(i)
Find the volume of one ball.
1 mark
(ii)
10 balls are painted with neon colours. Determine the area of painted surface.
1 mark
(iii)(a)
Find the volume of empty space in the box.
2 marks
OR
(iii)(b)
The lowermost layer of the balls covers the base of the box edge to edge when balls are placed evenly adjacent to each other. (A) How much area is covered by one ball? (B) How many balls are there in lowermost layer?
2 marks

Figure in the paper: Photograph of a cuboidal box filled with colourful balls, with a label on its front. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/4/1.

Hints

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

Hint 1

Each ball is a sphere. Take radius = half the diameter.

Hint 2

Empty space = volume of box minus total volume of the balls. Painted area uses the surface area of a sphere.

Hint 3

In the bottom layer each ball sits in a square of side 7 cm. How many such squares fit along 42 cm and along 91 cm?

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 $r = \frac{7}{2}$ cm. Volume of one ball $= \frac{4}{3} π r^{3} = \frac{4}{3} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times \frac{7}{2} = \frac{539}{3} \approx 179.67 cm^{3}$ .
1 mark
(ii) Surface area of 10 balls $= 10 \times 4 π r^{2} = 10 \times 4 \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} = 10 \times 154 = 1540 cm^{2}$ .
1 mark
(iii)(a) Volume of box $= l \times b \times h = 42 \times 91 \times 50 = 191100 cm^{3}$ .
½ mark
(iii)(a) Volume of 600 balls $= 600 \times \frac{539}{3} = 107800 cm^{3}$ .
½ mark
(iii)(a) Volume of empty space $= 191100 - 107800 = 83300 cm^{3}$ .
1 mark
(iii)(b)(A) Balls placed edge to edge each take up a square of side equal to the diameter, 7 cm. Area covered by one ball $= 7 \times 7 = 49 cm^{2}$ .
1 mark
(iii)(b)(B) Balls along the length $= \frac{42}{7} = 6$ and along the breadth $= \frac{91}{7} = 13$ . Number of balls in the lowermost layer $= 6 \times 13 = 78$ . (Check: $\frac{42 \times 91}{49} = 78$ .)
1 mark

Answer: (i) $\frac{539}{3} \approx 179.67$ cm³; (ii) 1540 cm²; (iii)(a) 83300 cm³; (iii)(b) (A) 49 cm², (B) 78 balls

Where the marks are

Uses $\frac{4}{3} π r^{3}$ with $r = 3.5$ cm to get $\frac{539}{3}$ cm³ (1 mark)
Uses $10 \times 4 π r^{2}$ to get 1540 cm² (1 mark)
Volume of box is 191100 cm³ (½ mark)
Volume of 600 balls is 107800 cm³ (½ mark)
Empty space is 83300 cm³ (1 mark)
Area covered by one ball is $7 \times 7 = 49$ cm² (1 mark)
Number of balls in the bottom layer is 78 (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 paper2025 board paper, set 430/4/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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