Hint
The boundary includes two radii. Substitute the resulting angle into the area formula, then complete the square.
Worked solution
(a)
From \(\displaystyle 2r+ r\theta = 24, \theta = \frac{24}{r}- 2\). Then \(\displaystyle A= (\frac{1}{2})r^{2}(\frac{24}{r}- 2)= 12r- r^{2}\).
Answer: \(\displaystyle \theta = \frac{24}{r}- 2\); \(A=12r-r^2\).
(b)
\(\displaystyle A=36-(r-6)^{2}\). The square is nonnegative, so no area exceeds \(\displaystyle 36 \mathrm{cm}^{2}\); equality occurs at \(\displaystyle r=6 \mathrm{cm}\).
Answer: Maximum \(36\text{ cm}^2\), at \(\displaystyle r=6 \mathrm{cm}\).
(c)
At \(\displaystyle r= 6, \theta = \frac{24}{6}- 2= 2\) rad. The radius is positive and 0<2<2π, so the algebraic maximum is attained by a proper sector.
Answer: \(\theta=2\text{ rad}\), with r>0 and \(0<2<2\pi\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Use the complete-perimeter equation \(\displaystyle 2r+r\theta =24\).
- 1 mark: Obtain \(\displaystyle \theta = \frac{24}{r}- 2\).
- 1 mark: Derive \(\displaystyle A=12r-r^{2}\).
Alternative methods: Eliminating arc length first is accepted.
Common error: Setting the arc length alone equal to 24.
(b) · 3 marks
- 1 mark: Use a valid quadratic maximum method.
- 1 mark: Obtain the corresponding radius 6 cm.
- 1 mark: Obtain maximum area \(\displaystyle 36 \mathrm{cm}^{2}\).
Alternative methods: Differentiation with a maximum justification is accepted.
Common error: Finding the vertex radius but reporting it as the maximum area.
(c) · 2 marks
- 1 mark: Calculate \(\displaystyle \theta =2\) rad.
- 1 mark: Check positive radius and a proper-sector angle strictly between 0 and 2π.
Alternative methods: An equivalent explicit domain check is accepted.
Common error: Assuming every algebraic vertex is a valid geometric region.
Total: 8 marks.