Hint
Angles in \(\displaystyle r\theta\) and \(\displaystyle \frac{r^2\theta}{2}\) must be radians. Use \(\displaystyle r\theta\) and remove the requested factor π.
Worked solution
(a)
\(s = 3 \pi\) and \(A = 9 \pi\) use the radian angle.
Answer: \(3 \pi\) cm; \(9 \pi\) \(\displaystyle \mathrm{cm}^{2}\)
(b)
Add the arc and the two straight radii.
Answer: \(3 \pi + 12\) cm
(c)
Add both radii for the sector boundary.
\(12\)
Answer: \(12\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Use the radian formulas \(\displaystyle r\theta\) and \(\displaystyle \frac{r^2\theta}{2}\) with the correct units.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not substitute degrees directly into radian formulas.
(b) · 2 marks
- 1 mark: Add the arc and both straight radii.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A sector perimeter is not just its arc.
(c) · 2 marks
- 1 mark: Add both radii for the sector boundary.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not substitute degrees directly into radian formulas.
Total: 6 marks.