MathsCBSE Class 10 · Standard · Board exam 2024, set 30/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2024: question 36

Set in the CBSE Class 10 board exam of 2024, paper 30/3/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedAreas Related to Circles

A stable owner has four horses. He usually tie these horses with 7 m long rope to pegs at each corner of a square shaped grass field of 20 m length, to graze in his farm. But tying with rope sometimes results in injuries to his horses, so he decided to build fence around the area so that each horse can graze.

The figure shows a square field ABCD with side 20 m, with a horse tied at each corner by a rope of length 7 m, grazing in a quarter circle.

Based on the above, answer the following questions:

(i)
Find the area of the square shaped grass field.
1 mark
(ii)
(a) Find the area of the total field in which these horses can graze.
2 marks
OR
(ii)
(b) If the length of the rope of each horse is increased from 7 m to 10 m, find the area grazed by one horse. (Use $π = 3.14$ )
2 marks
(iii)
What is area of the field that is left ungrazed, if the length of the rope of each horse is 7 cm?
1 mark

Figure in the paper: Square field ABCD with A at top left, D at top right, B at bottom left and C at bottom right. Side AB is marked 20 m and the top side AD has a 7 m arrow marked from A. Each corner has a horse tied with a rope drawn as a quarter-circle arc of radius 7 m inside the square. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/3/1.

Hints

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

Hint 1

Each horse is tied at a corner of the square, so it can graze a region shaped like a quarter circle.

Hint 2

The area of a quadrant is $\frac{1}{4} π r^{2}$ . Four quadrants of the same radius together make one full circle.

Hint 3

In (iii), take care with units: convert the grazed area from cm² to m² before subtracting it from the field'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) Area of the square field $= (side)^{2} = 20 \times 20 = 400 m^{2}$
1 mark
(ii)(a) Each horse grazes a quadrant of radius 7 m (the corner angle is $9 0^{\circ}$ ). So the total grazed area $= 4 \times \frac{1}{4} π r^{2} = 4 \times \frac{1}{4} \times \frac{22}{7} \times 7 \times 7$
1 mark
(ii)(a) $= 154 m^{2}$
1 mark
(ii)(b) With a 10 m rope, area grazed by one horse $= \frac{1}{4} π r^{2} = \frac{1}{4} \times 3.14 \times 10 \times 10$
1 mark
(ii)(b) $= 78.5 m^{2}$
1 mark
(iii) The rope is 7 cm $= 0.07$ m. Area grazed by the four horses $= 4 \times \frac{1}{4} \times \frac{22}{7} \times 7 \times 7 = 154 cm^{2} = 0.0154 m^{2}$
½ mark
(iii) Ungrazed area $= 400 - 0.0154 = 399.9846 m^{2}$
½ mark

Answer: (i) 400 m²; (ii)(a) 154 m², (ii)(b) 78.5 m²; (iii) 399.9846 m²

Where the marks are

(i) Area 400 m² (1 mark)
(ii)(a) Writes 4 quadrants of radius 7 m (1 mark) and gets 154 m² (1 mark)
(ii)(b) Writes a quadrant of radius 10 m with π = 3.14 (1 mark) and gets 78.5 m² (1 mark)
(iii) Grazed area 154 cm² = 0.0154 m² (½ mark); ungrazed area 399.9846 m² (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Sector of a Circle

Practise this chapterBoard-style questions from Areas Related to Circles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/3/1 (Standard): 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 Standard 2024: question 36

Hints

Step-by-step solution

Revise the chapter

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