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?}
Show 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.
Step 1: Understanding the Concept:
The silver wire is used for the circular boundary as well as the 5 internal diameters. To find the area of a sector, we divide the total area of the circle by the number of equal sectors (10). Step 2: Key Formula or Approach:
1. Circumference = $\pi d$.
2. Total wire = Circumference + $5 \times d$.
3. Area of sector = $\frac{\pi r^2}{10}$. Step 3: Detailed Explanation:
1. (i) Radius: $r = \frac{35}{2} = 17.5$ mm.
2. (ii) Circumference: $C = \pi \times 35 = \frac{22}{7} \times 35 = 110$ mm.
3. (iii) (a) Total Wire:
- $\text{Wire} = \text{Circumference} + (5 \times \text{Diameter})$
- $\text{Wire} = 110 + (5 \times 35) = 110 + 175 = 285$ mm.
4. (iii) (b) OR Area of each sector:
- $\text{Total Area} = \pi r^2 = \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2} = \frac{11 \times 5 \times 35}{2} = \frac{1925}{2} = 962.5$ mm$^2$.
- $\text{Sector Area} = \frac{962.5}{10} = 96.25$ mm$^2$. Step 4: Final Answer:
(i) 17.5 mm. (ii) 110 mm. (iii)(a) 285 mm or (iii)(b) 96.25 mm$^2$.
