Question

. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in Fig. 12.12. Find:
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.

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Answer 1

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⭐ A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find :

• Total length of silver wire required
• The Area of Each sector of the brooch

$\underline{\underline{\sf{\maltese\:\:Given}}}$

• Diameter of Brooch = 35 mm

• The wire is also used in making 5 diameters

• Wire divides the circle into 10 equal sectors

$\underline{\underline{\sf{\maltese\:\:To\:Find}}}$

• Total length of silver wire required
• The Area of Each sector of the brooch

$\underline{\underline{\sf{\maltese\:\:Answer}}}$

• Total length of silver wire required = 285 mm

• Area of Each sector of the brooch = 96.25 mm²

$\underline{\underline{\sf{\maltese\:\:Calculations}}}$

Before going into calculation part, we need to know about some basic terms related to the answer

• Diameter :  The diameter is the length of the line through the center that touches two points on the edge of the circle.
• Radius :  A line segment extending from the center of a circle or sphere to the circumference

Also :

• Diameter = 2 × Radius
• Radius = Diameter

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First Let's Find

• Total length of silver wire required

Total length of silver wire required

= Length of Wire in Circle + Wire Used in 5 diameters

⇒ Total length of silver wire required

= Circumference of Circle + Wire Used in 5 diameters

⇒ Total length of silver wire required = 2πr + Wire Used in 5 diameters

⇒ Total length of silver wire required = 2πr + (5 × Diameter of Circle)

⇒ Total length of silver wire required = 2 × π × r + (5 × Diameter of Circle)

⇒ Total length of silver wire required

= 2 × π × (diameter/2) + (5 × Diameter of Circle)

⇒ Total length of silver wire required

= 2 × π × (35/2 mm) + (5 × Diameter of Circle)

⇒ Total length of silver wire required = 2 × π × (35/2 mm) + (5 × 35 mm)

⇒ Total length of silver wire required = 2 × π × (35/2 mm) + (175 mm)

Since π = 22/7

⇒ Total length of silver wire required = 2 × 22/7 × (35/2 mm) + (175 mm)

⇒ Total length of silver wire required = 2 × 22/7 × 35/2 mm + 175 mm

⇒ Total length of silver wire required = 2 × 22/1 × 5/2 mm + 175 mm

⇒ Total length of silver wire required = 2 × 22 × 5/2 mm + 175 mm

⇒ Total length of silver wire required = 22× 5/1 mm + 175 mm

⇒ Total length of silver wire required = 22 × 5 mm + 175 mm

⇒ Total length of silver wire required = 110 mm + 175 mm

⇒ Total length of silver wire required = 285 mm

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Now Let's Find

• The Area of Each sector of the brooch

Area of Each sector of the brooch = Area of Circle/10

⇒ Area of Each sector of the brooch = πr²/10

⇒ Area of Each sector of the brooch = π × r²/10

⇒ Area of Each sector of the brooch = π × (diameter/2)²/10

Since π = 22/7

⇒ Area of Each sector of the brooch = 22/7 × (diameter/2)²/10

⇒ Area of Each sector of the brooch = 22/7 × (35/2 mm)²/10

⇒ Area of Each sector of the brooch = 22/7 × ((35 × 35/(2 × 2)) mm²/10

⇒ Area of Each sector of the brooch = 22/1 × ((5 × 35/(2 × 2))  mm²/10

⇒ Area of Each sector of the brooch = 22 × ((5 × 35/(2 × 2))  mm²/10

⇒ Area of Each sector of the brooch = 22 × ((5 × 35)/(2 × 2)) mm²/10

⇒ Area of Each sector of the brooch = 11 × ((5 × 35)/(1 × 2)) mm²/10

⇒ Area of Each sector of the brooch = 11 × ((5 × 35)/2) mm²/10

⇒ Area of Each sector of the brooch = 11 × (175/2) mm²/10

⇒ Area of Each sector of the brooch = 11 × (175)/(2 × 10) mm²

⇒ Area of Each sector of the brooch = 11 × 175/20  mm²

⇒ Area of Each sector of the brooch = 11 × 8.75 mm²

⇒ Area of Each sector of the brooch = 96.25 mm²

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