C2 January 2005 Q7
7.
Figure 1 shows the triangle \(ABC\), with \(AB = 8\) cm, \(AC = 11\) cm and \(\angle BAC = 0.7\) radians. The arc \(BD\), where \(D\) lies on \(AC\), is an arc of a circle with centre \(A\) and radius 8 cm. The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(BC\) and \(CD\) and the arc \(BD\).
Find
(a) the length of the arc \(BD\), (2)
(b) the perimeter of \(R\), giving your answer to 3 significant figures, (4)
(c) the area of \(R\), giving your answer to 3 significant figures. (5)
| Scheme | Marks |
|---|---|
| \(r\theta = 8 \times 0.7, = 5.6\ (cm)\) | M1, A1 |
| (2) |
Notes
M1 for quoting and attempting to use correct formula
| Scheme | Marks |
|---|---|
| \(BC^2 = 8^2 + 11^2 - 2 \times 8 \times 11 \times \cos 0.7\) | M1 |
| \(\Rightarrow BC = 7.098\) or 7.10 (Awrt) or \(\sqrt{(50.4)}\) or better | A1 |
| Perimeter \(= (a) + (11 - 8) + BC, = 15.7\ (cm)\) | M1, A1cao |
| (4) |
Notes
1st M1 for attempting to use cosine rule (formula given)
| Scheme | Marks |
|---|---|
| \(\Delta = \dfrac{1}{2}ab\sin c =, \dfrac{1}{2} \times 11 \times 8 \times \sin 0.7\) | M1, A1 |
| Sector \(= \dfrac{1}{2}r^2\theta =, \dfrac{1}{2} \times 8^2 \times 0.7\) | M1, A1 |
| Area of \(R = 28.345\ldots\ldots - 22.4 = 5.9455 = 5.95\ (cm^2)\) | A1 |
| (5) | |
| (11 marks) |
Notes
Final A1 accept 3sf or better
M1 for quoting and attempting to use correct formula