An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight line with AB = 5 cm and BD = 4 cm. Angle BAC = 0.6 radians and AC is the longest side of the triangle ABC. (a) Show that angle ABC = 1.76 radians, correct to 3 significant figures. (b) Find the area of the emblem.

A-Level Edexcel Maths Pure Solving Equations: An emblem, as shown in Figure 1,

An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 4

Question 5

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An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie on a straight... show full transcript

Worked Solution & Example Answer:An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 4

Step 1

Show that angle ABC = 1.76 radians, correct to 3 significant figures.

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Answer

To find angle ABC, we can use the sine rule in triangle ABC. We know the length of sides AB and AC, as well as angle BAC.

Using the sine rule: $\frac{AB}{\sin(ABC)} = \frac{AC}{\sin(0.6)}$

Letting AC be the longest side, we can first find its length using the law of cosines: $AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(0.6)$ Substituting the known values:
$AC^2 = 5^2 + 4^2 - 2 \cdot 5 \cdot 4 \cdot \cos(0.6)$
$AC^2 = 25 + 16 - 40 \cdot 0.8256 \approx 25 + 16 - 33.024 = 7.976$
$AC = \sqrt{7.976} \approx 2.827$

Then substituting into the sine rule gives: $\frac{5}{\sin(ABC)} = \frac{2.827}{\sin(0.6)}$ Now we solve for (\sin(ABC)): $\sin(ABC) = \frac{5 \cdot \sin(0.6)}{2.827} \approx \frac{5 \cdot 0.5736}{2.827} \approx 1.0147$ Finally, we find angle ABC: $ABC = \arcsin(1.0147) \approx 1.76 \text{ radians}$ Thus, angle ABC is approximately 1.76 radians, correct to 3 significant figures.

Step 2

Find the area of the emblem.

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Answer

The area of the emblem consists of the area of triangle ABC and the area of sector CBD.
For the area of triangle ABC, we can use the formula: $\text{Area} = \frac{1}{2} \cdot AB \cdot AC \cdot \sin(0.6)$
Using the values obtained: $\text{Area}_{ABC} = \frac{1}{2} \cdot 5 \cdot 2.827 \cdot \sin(0.6) \approx \frac{1}{2} \cdot 5 \cdot 2.827 \cdot 0.5736 \approx 4.063$

Next, we calculate the area of sector CBD: $\text{Area}_{sector} = \frac{1}{2} \cdot r^2 \cdot \theta$
Where r is the radius (4 cm) and (\theta) is the angle in radians (1.76 radians): $\text{Area}_{sector} = \frac{1}{2} \cdot 4^2 \cdot (1.76) \approx \frac{1}{2} \cdot 16 \cdot 1.76 = 14.08$

Adding both areas together gives: $\text{Total Area} = \text{Area}_{ABC} + \text{Area}_{sector} \approx 4.063 + 14.08 = 18.143$ Thus, the area of the emblem is approximately 18.14 cm².

An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 4

Worked Solution & Example Answer:An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 4

Show that angle ABC = 1.76 radians, correct to 3 significant figures.

Find the area of the emblem.

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