IB Mathematics SL 3.4 length of an arc area of a sector AI SL Paper 1- Exam Style Questions- New Syllabus

(a)
The shaded region is a sector of an annulus. The area of the full annulus is:
\( \pi R^2 – \pi r^2 = \pi(4^2 – 2.8^2) = \pi(16 – 7.84) = 8.16\pi \ \text{m}^2 \).
Since the circle is divided into 5 equal sectors:
Area of one sector \( = \frac{1}{5} \times 8.16\pi \approx 5.127 \ \text{m}^2 \).
Since \( 5.127 < 6 \), Matt has enough dye.
Area \( \approx 5.13 \ \text{m}^2 \); yes.

(b)
The path consists of:
1. Outer arc through three sectors: angle \( = \frac{3}{5} \times 360^\circ = 216^\circ \)
Arc length \( = \frac{216}{360} \times 2\pi \times 4 = \frac{3}{5} \times 8\pi \approx 15.08 \ \text{m} \).
2. Return along a radial path of length \(4 \ \text{m} \).
Total distance \( \approx 15.08 + 4 = 19.08 \ \text{m} \).
\( \approx 19.1 \ \text{m} \).