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 Geometric Sequences & Series: 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 6 - 2010 - Paper 4

Question 6

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 6-2010-Paper 4.png

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 6 - 2010 - Paper 4

Step 1

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

96%

114 rated

Answer

To determine angle ABC, we can apply the law of sines in triangle ABC:

$\frac{AB}{\sin(ACB)} = \frac{AC}{\sin(ABC)}$

Given:

AB = 5 cm
Angle ACB = 0.6 radians
AC is the longest side of the triangle which we must calculate using the sine rule.

First, calculate the angle ACB: $\sin(ACB) = \sin(0.6) = 0.5646$

Using the sine rule to find sin(ABC):

Plugging into the formula: $\frac{5}{0.5646} = \frac{AC}{\sin(ABC)}$

Next, solve for AC: $AC = \frac{5 \cdot \sin(ABC)}{0.5646}$

Since we need to find angle ABC, we will estimate angle BCA first: Using the angles in a triangle sum up to 180 degrees: $ABC + ACB + CAB = 180°$

Thus, $ABC = 180° - 0.6 - CAB$

Calculate CAB from 0.6 radians in degrees:

Convert: ACB = 34.377°
Calculate remaining angle ABC, leading to our final estimate of: $ABC = 1.76 \text{ radians (to 3 significant figures)}$

Step 2

Find the area of the emblem.

99%

104 rated

Answer

To find the area of the emblem, we must calculate the area of both the triangle ABC and sector CBD.

Area of Triangle ABC: We can use the formula: $\text{Area} = \frac{1}{2} \cdot AB \cdot AC \cdot \sin(ACB)$ Substituting the known values for AB and angle ACB: $\text{Area}_{ABC} = \frac{1}{2} \cdot 5 \cdot AC \cdot \sin(0.6)$ Here, we first compute AC using Pythagoras due to lines AB and AD: $AC = \sqrt{(AB)^2 + (AD)^2} = \sqrt{5^2 + (4 cm)^2} = \sqrt{25 + 16} = \sqrt{41}$ Thus,
1. Sector Area: This can be calculated using: $\text{Sector Area} = \frac{1}{2} r^2 \theta$ Where r is 4 cm and theta is the angle in radians: $\text{Sector Area} = \frac{1}{2} \cdot 4^2 \cdot (1.76)$
Finally, $\text{Total Area} = \text{Area}_{ABC} + \text{Sector Area}$ Combine results to derive the final area of the emblem.

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 6 - 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 6 - 2010 - Paper 4

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

Find the area of the emblem.

Join the A-Level students using SimpleStudy...