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The shape shown in Figure 1 is a pattern for a pendant - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2
Question 7
The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle centre O, of radius 6 cm, and angle AOB = \( \frac{\pi}{3} \). The ci... show full transcript
Worked Solution & Example Answer:The shape shown in Figure 1 is a pattern for a pendant - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 2
Step 1
Step 2
(b) the radius of the circle C
Answer
For the circle C that touches the straight edges OA and OB, we need to use trigonometric properties. From the geometry of the sector,
Let ( r ) be the radius of circle C. The height from O to the tangent where circle C touches OA and OB can be described as:
\Rightarrow r = \frac{6 - r}{\sqrt{3}}$$ Solving for \( r \): Multiplying both sides by \( \sqrt{3} (6 - r) \): $$r\sqrt{3} + r = 6\sqrt{3}$$ Thus, $$r(1 + \sqrt{3}) = 6\sqrt{3}\n r = \frac{6\sqrt{3}}{1 + \sqrt{3}} = 2\text{ cm}$$.Step 3
(c) Find the area of the shaded region
Answer
To find the area of the shaded region, we first calculate the area of the sector OAB, which is ( 12\pi \text{ cm}^2 ), and then subtract the area of circle C:
The area of circle C is:
Thus, the area of the shaded region is:
The shaded area is approximately ( 8\pi \approx 25.1 \text{ cm}^2 ) or exactly ( 8\pi \text{ cm}^2 ).