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 as shown in figure. Based on the above information, answer the following questions :
Question: 1
What is the radius of circle ?
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Always double-check the units given in the question. In some versions of this problem, it might be mm or cm. Ensure you maintain consistency.
Step 1: Understanding the Concept:
The radius of a circle is half of its diameter. Step 2: Key Formula or Approach:
\[ r = \frac{d}{2} \] Step 3: Detailed Explanation:
The given diameter (\(d\)) of the circular brooch is 35 mm.
Using the formula for radius:
\[ r = \frac{35}{2} \text{ mm} \]
\[ r = 17.5 \text{ mm} \] Step 4: Final Answer:
The radius of the circle is 17.5 mm.
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Question: 2
What is the circumference of the brooch ?
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If the diameter is a multiple of 7, using \( \pi = \frac{22}{7} \) simplifies calculations significantly.
Step 1: Understanding the Concept:
The circumference of a circle is the total distance around the edge of the circle. Step 2: Key Formula or Approach:
\[ C = \pi d \quad \text{or} \quad C = 2\pi r \]
Take \( \pi = \frac{22}{7} \). Step 3: Detailed Explanation:
Diameter \(d = 35\) mm.
\[ C = \frac{22}{7} \times 35 \]
\[ C = 22 \times 5 \]
\[ C = 110 \text{ mm} \] Step 4: Final Answer:
The circumference of the brooch is 110 mm.
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Question: 3
What is the total length of silver wire required ?
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Don't forget to add the circumference! Many students only calculate the diameter lengths.
Step 1: Understanding the Concept:
The total wire length consists of the length used for the circular boundary plus the length used for the internal diameters. Step 2: Key Formula or Approach:
\[ \text{Total Length} = \text{Circumference} + (5 \times \text{Diameter}) \] Step 3: Detailed Explanation:
Length used for circumference \( = 110 \text{ mm} \) (as calculated in part ii).
Length used for 5 diameters \( = 5 \times 35 \text{ mm} = 175 \text{ mm} \).
Total wire required:
\[ L = 110 + 175 \]
\[ L = 285 \text{ mm} \] Step 4: Final Answer:
The total length of silver wire required is 285 mm.
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Question: 4
What is the area of each sector of the brooch ?
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The central angle of each sector is \( \frac{360^{\circ}}{10} = 36^{\circ} \). You can use the sector formula \( \frac{\theta}{360} \times \pi r^2 \), which will yield the same result.
Step 1: Understanding the Concept:
The circle is divided into 10 equal sectors. The area of each sector is one-tenth of the total area of the circle. Step 2: Key Formula or Approach:
\[ \text{Area of sector} = \frac{\pi r^2}{10} \] Step 3: Detailed Explanation:
Total Area of Circle \( = \pi r^2 \).
\[ \text{Area} = \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2} \]
\[ \text{Area} = \frac{11 \times 5 \times 35}{2} \]
\[ \text{Area} = \frac{1925}{2} = 962.5 \text{ mm}^2 \]
Since there are 10 equal sectors:
\[ \text{Area of each sector} = \frac{962.5}{10} \]
\[ \text{Area of each sector} = 96.25 \text{ mm}^2 \] Step 4: Final Answer:
The area of each sector of the brooch is 96.25 \text{ mm}\^2.
