Question
The diagram shows a circle, centre O, with radius 4 cm. Points A and B lie on the circumference of the circle and AÔB = θ , where 0 ≤ θ ≤ \(\pi \).
Find the area of the shaded region, in terms of θ.
The area of the shaded region is 12 cm2. Find the value of θ.
Answer/Explanation
Markscheme
valid approach to find area of segment (M1)
eg area of sector – area of triangle, \(\frac{1}{2}{r^2}\left( {\theta – {\text{sin}}\theta } \right)\)
correct substitution (A1)
eg \(\frac{1}{4}{\left( 4 \right)^2}\theta – \frac{1}{2}{\left( 4 \right)^2}{\text{sin}}\theta ,\,\,\frac{1}{2} \times 16\left[ {\theta – {\text{sin}}\theta } \right]\)
area = 80 – 8 sinθ, 8(θ – sinθ) A1 N2
[3 marks]
setting their area expression equal to 12 (M1)
eg 12 = 8(θ – sinθ)
2.26717
θ = 2.27 (do not accept an answer in degrees) A2 N3
[3 marks]