Question 3

We can find the arc length using a standard formula.

Step 1 — Calculate arc length for part (i)

Let's write down the given values. The radius $r$ is $\mathbf{3.5\text{ cm}}$. The angle $\theta$ at the centre is $\mathbf{60^\circ}$. We use the formula for arc length $L = \frac{\theta}{360^\circ} \times 2\pi r$.

$L = \frac{60}{360} \times 2 \times \frac{22}{7} \times 3.5$

$= \frac{1}{6} \times 2 \times \frac{22}{7} \times \frac{7}{2}$

$= \frac{1}{6} \times 22$

$= \frac{11}{3}$

$= 3.666...$

$\boxed{3.67 \text{ cm}}$

Step 2 — Calculate arc length for part (ii)

Let's write down the given values. The radius $r$ is $\mathbf{6.3\text{ m}}$. The angle $\theta$ at the centre is $\mathbf{120^\circ}$. We use the formula for arc length $L = \frac{\theta}{360^\circ} \times 2\pi r$.

$L = \frac{120}{360} \times 2 \times \frac{22}{7} \times 6.3$

$= \frac{1}{3} \times 2 \times \frac{22}{7} \times \frac{63}{10}$

$= \frac{1}{3} \times 2 \times 22 \times \frac{9}{10}$

$= \frac{1}{3} \times 44 \times \frac{9}{10}$

$= 44 \times \frac{3}{10}$

$= \frac{132}{10}$

$\boxed{13.2 \text{ m}}$

Answer

(i) The length of the arc is $3.67 \text{ cm}$. (ii) The length of the arc is $13.2 \text{ m}$.