Question

A brooch is crafted from silver wire in the shape of a circle with a diameter of 35 mm. The wire is also used to create 5 diameters, dividing the circle into 10 equal sectors.

(i) What is the radius of circle?
(ii) What is the circumference of the brooch?
(iii) (a) What is the total length of silver wire required? OR
(iii) (b) What is the area of each sector of the brooch?}

Hint:

For problems with multiple diameters, check the wording carefully. "5 diameters" means 10 radii, which is exactly what divides a circle into 10 sectors.

Answer 1

A brooch is made from silver wire in the shape of a circle, diameter = 35 mm.
The wire is also used to make 5 diameters, dividing the circle into 10 equal sectors.

(i) Radius of the circle
Diameter = 35 mm 
\[ \text{Radius} = \frac{35}{2} = 17.5\ \text{mm} \]

(ii) Circumference of the brooch
Circumference formula: \[ C = 2\pi r = 2\pi(17.5) \] \[ = 35\pi\ \text{mm} \] Using \(\pi = 22/7\): \[ C = 35 \times \frac{22}{7} = 110\ \text{mm} \]

(iii) (a) Total length of silver wire required
Wire is used for:

• The circumference
• 5 diameters

Length of 1 diameter: 35 mm 
Total length of 5 diameters: \[ 5 \times 35 = 175\ \text{mm} \]

Total silver wire required: \[ 110 + 175 = 285\ \text{mm} \]

 

(iii) (b) Area of each sector of the brooch
Area of circle: \[ A = \pi r^2 = \pi (17.5)^2 = \pi (306.25) \] With \(\pi = 22/7\): \[ A = \frac{22}{7} \times 306.25 = 962.5\ \text{mm}^2 \] There are 10 sectors, so area of each sector: \[ \text{Sector area} = \frac{962.5}{10} = 96.25\ \text{mm}^2 \]

Final Answers:
(i) Radius = 17.5 mm
(ii) Circumference = 110 mm
(iii)(a) Total wire length required = 285 mm
(iii)(b) Area of each sector = 96.25 mm²