Q) A brooch is a small piece of jewellery which has a pin at the back so it can be fastened on a dress, blouse or coat. Designs of some brooch are shown below. Observe them carefully.
Design A: Brooch A is made with silver wire in the form of a circle with diameter 28mm. The wire used for making 4 diameters which divide the circle into 8 equal parts.
Design B: Brooch b is made two colours_ Gold and silver. Outer part is made with Gold. The circumference of silver part is 44mm and the gold part is 3mm wide everywhere.
Refer to Design A
1. The total length of silver wire required is
a) 180 mm b) 200 mm c) 250 mm d) 280 mm
2. The area of each sector of the brooch is
a) 44 mm2 b) 52 mm2 c) 77 mm2 d) 68 mm2
Refer to Design B
3. The circumference of outer part (golden) is
a) 48.49 mm b) 82.2 mm c) 72.50 mm d) 62.86 mm
4. The difference of areas of golden and silver parts is
a) 18 b) 44 c) 51 d) 64
5. A boy is playing with brooch B. He makes revolution with it along its edge. How many complete revolutions must it take to cover 80 π mm ?
a) 2 b) 3 c) 4 d) 5
Ans:
1. Design A – Length of wire:
Step 1: ∵ given diameter of circle in design A = 28 mm 
∴ Radius of circle in design A = 
= 14 mm
Step 2: Length of wire used in design A = Circumference of Circle + 4 diameters
Here, Circumference = 2 π r = 2 
x 14 = 88 mm
Length of 4 diameters = 4 x 28 = 112 mm
∴ Length of wire used i design A = 88 + 112 = 200 mm
Therefore, option (b) is correct.
2. Design A – Area of a sector:
Step 3: Since Area of a circle:
= π r 2 = x (14)2
= 22 x 14 x 2 = 616 mm2
Step 4: Area of a sector = 
= 
= 77 mm2
Therefore, option (c) is correct.
3. Design B – Circumference of the outer part (golden):
Step 5: By the given information, the circumference of the silver part = 44 mm 
∵ Circumference = 2 π r
∴ r = 
∴ rsilver = 
= 7 mm
Step 6: Since, the outer radius of the Golden part = radius of the silver part + width 3 mm
= 7 + 3 = 10 mm
Step 7: ∴ Circumference of the Golden part = 2 x π x 10 = 20 π
= 20 x 
= 62.86 mm
Therefore, option (d) is correct.
4. Design B – Difference of areas of golden and silver parts:
Step 8: Difference Area of the Golden and Silver parts
= Outer Circle’s Area – Inner circle’s area
= π r outer 2 – π r inner 2
= π (10) 2 – π (7) 2
= 100 π – 49 π = 51 π mm 2
Therefore, option (c) is correct.
5. Number of revolutions by brooch:
Step 9: Length covered by the brooch in 1 revolution
= circumference of the outer part
= 20 π mm
(calculated in step 7)
Step 10: Total length to be covered = 80 π mm
Since this length is to be covered by the brooch in multiple revolutions
∴ Total length = length covered in one revolution x number of revolutions
∴ Number of revolutions = 
= 
= 4
Therefore, option (c) is correct.
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