MathsCBSE Class 10 · Standard · Board exam 2023, set 30/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2023: question 38

Set in the CBSE Class 10 board exam of 2023, paper 30/4/1, 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-basedAreas Related to Circles

Governing council of a local public development authority of Dehradun decided to build an adventurous playground on the top of a hill, which will have adequate space for parking.

After survey, it was decided to build rectangular playground, with a semi-circular area allotted for parking at one end of the playground. The length and breadth of the rectangular playground are 14 units and 7 units, respectively. There are two quadrants of radius 2 units on one side for special seats.

Based on the above information, answer the following questions:

(i)
What is the total perimeter of the parking area?
1 mark
(ii)(a)
What is the total area of parking and the two quadrants?
2 marks
OR
(ii)(b)
What is the ratio of area of playground to the area of parking area?
2 marks
(iii)
Find the cost of fencing the playground and parking area at the rate of ₹ 2 per unit.
1 mark

Figure in the paper: Rectangular playground with length 14 units (bottom side) and breadth 7 units (right side marked). Two quadrants of radius 2 units, labelled I (top left corner) and II (bottom left corner), drawn at the left corners of the rectangle. A semicircular region labelled PARKING AREA is attached to the right side of the rectangle, with its diameter equal to the 7-unit side. The rectangle is labelled PLAYGROUND. The boundary is drawn with a zigzag fence line. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/4/1.

Hints

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

Hint 1

The parking semicircle has the 7-unit breadth as its diameter.

Hint 2

The perimeter of a semicircle is the arc plus the diameter. A quadrant has one-fourth the area of a circle.

Hint 3

For fencing, take the full rectangle perimeter plus the semicircular arc, then multiply by ₹2.

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) The parking area is a semicircle on the 7-unit side, so its diameter is 7 and $r = \frac{7}{2}$ units.
Perimeter $= π r + 2 r = \frac{22}{7} \times \frac{7}{2} + 7 = 11 + 7 = 18$ units
1 mark
(ii)(a) Area of parking $= \frac{1}{2} π r^{2} = \frac{1}{2} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} = \frac{77}{4}$ sq. units
1 mark
(ii)(a) Area of two quadrants of radius 2 $= 2 \times \frac{1}{4} \times \frac{22}{7} \times 2 \times 2 = \frac{44}{7}$ sq. units
½ mark
(ii)(a) Total area $= \frac{77}{4} + \frac{44}{7} = \frac{539 + 176}{28} = \frac{715}{28} \approx 25.54$ sq. units
½ mark
(ii)(b) Area of playground $= 14 \times 7 = 98$ sq. units.
Area of parking $= \frac{77}{4}$ sq. units.
1 mark
(ii)(b) Ratio $= \frac{98}{77/4} = \frac{392}{77} = \frac{56}{11}$
So the ratio of playground area to parking area is $56 : 11$ .
1 mark
(iii) Fence the whole rectangle and the curved edge of the parking area:
Length $= 2 (l + b) + π r = 2 (14 + 7) + \frac{22}{7} \times \frac{7}{2} = 42 + 11 = 53$ units
½ mark
(iii) Cost of fencing $= 53 \times ₹2 = ₹106$
½ mark

Answer: (i) 18 units; (ii)(a) 715/28 ≈ 25.54 sq. units; (ii)(b) 56 : 11; (iii) ₹106

Where the marks are

(i) Perimeter $π r + 2 r = 18$ units (1 mark)
(ii)(a) Area of semicircle $= 77/4$ (1 mark)
(ii)(a) Area of two quadrants $= 44/7$ (½ mark)
(ii)(a) Total $715/28 \approx 25.54$ sq. units (½ mark)
(ii)(b) Areas 98 and 77/4 (1 mark)
(ii)(b) Ratio 56 : 11 (1 mark)
(iii) Fencing length 53 units (½ mark)
(iii) Cost ₹106 (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)

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 paper2023 board paper, set 30/4/1 (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 2023: question 38

Hints

Step-by-step solution

Revise the chapter

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