IB Mathematics SL 3.4 The circle radian measure of angles AA SL Paper 2- Exam Style Questions- New Syllabus
Question
Most-appropriate topic codes (Mathematics: analysis and approaches guide):
• SL 3.4: The circle: radian measure of angles; length of an arc; area of a sector — Part a, b
• Prior learning: The circle, its centre and radius, area and circumference — Part a, b
▶️ Answer/Explanation
(a) Calculate the sector area:
The area of a sector is given by \( A = \frac{1}{2}R^2\theta \), where \( \theta \) is in radians
Convert the angle to radians: \( 210^\circ = 210 \times \frac{\pi}{180} = \frac{7\pi}{6} \) radians.
\( A = \frac{1}{2} \times (19.5)^2 \times \frac{7\pi}{6} \)
\( A = \frac{1}{2} \times 380.25 \times \frac{7\pi}{6} = \frac{2661.75\pi}{12} \)
\( A = \frac{3549\pi}{16} \approx 697 \text{ cm}^2 \) (to 3 significant figures).
Answer: \( \boxed{697 \ \text{cm}^2} \)
(b) Determine the cone radius:
When the sector is folded into a cone, the arc length \( L \) of the sector becomes the circumference of the cone’s base.
Calculate the arc length using \( L = R\theta \)[cite: 1143]:
\( L = 19.5 \times \frac{7\pi}{6} = \frac{136.5\pi}{6} = \frac{91\pi}{4} \text{ cm} \).
Let \( r \) be the radius of the cone. The base circumference is \( 2\pi r \)[cite: 723].
\( 2\pi r = \frac{91\pi}{4} \)
Divide both sides by \( 2\pi \):
\( r = \frac{91}{8} = 11.375 \text{ cm} \).
Answer: \( \boxed{11.4 \ \text{cm}} \) (to 3 significant figures).