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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

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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

Find (a) the area of the sector OAB

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Answer

To find the area of the sector OAB, we use the formula:

Area=12r2θ\text{Area} = \frac{1}{2} r^2 \theta

where ( r = 6 ) cm and ( \theta = \frac{\pi}{3} ).

Calculating,

Area=12×(62)×π3=12×36×π3=6π cm2\text{Area} = \frac{1}{2} \times (6^2) \times \frac{\pi}{3} = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 6\pi \text{ cm}^2.

Step 2

Find (b) the radius of the circle C

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Answer

To find the radius ( r ) of the circle C that touches the two straight edges and arc AB, we can use a relationship derived from the geometry of the figure:

By trigonometry, the radius is given as:

sin⁡(30∘)=r6−r\sin(30^\circ) = \frac{r}{6 - r}

Here, substituting ( \sin(30^\circ) = \frac{1}{2} ), we have:

12=r6−r\frac{1}{2} = \frac{r}{6 - r}

From this, we can solve:

6−r=2r6 - r = 2r
6=3r⇒r=2 cm6 = 3r \Rightarrow r = 2 \text{ cm}.

Step 3

Find (c) the area of the shaded region

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Answer

To find the area of the shaded region, we subtract the area of circle C from the area of the sector OAB:

First, calculate the area of circle C:

Area of Circle C=πr2=π(2)2=4π cm2\text{Area of Circle C} = \pi r^2 = \pi (2)^2 = 4\pi \text{ cm}^2

Thus, the area of the shaded region is:

Area of the Shaded Region=Area of Sector OAB−Area of Circle C\text{Area of the Shaded Region} = \text{Area of Sector OAB} - \text{Area of Circle C}

Substituting the values:

=6π−4π=2π cm2= 6\pi - 4\pi = 2\pi \text{ cm}^2.

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