Arc lengths and sectors

(a) $88.4\text{ cm}^2$

1. Radius $= 30 \div 2 = 15$ cm.
   $$\pi \times 15^2 = 225\pi = 706.85\ldots$$
2. Each slice is a sector with angle $45^\circ$, an eighth of the circle.
   $$706.85\ldots \div 8 = 88.35\ldots$$

• P1 The whole area with radius 15.
• P1 Dividing by 8 (or multiplying by $\frac{45}{360}$).
• A1 $88.4\text{ cm}^2$ (awrt 88.4).

(a) $10\pi$

1. $$150/360\times 2\pi \times 12$$
2. Arc length is the fraction angle/360 of the full circumference.
3. Arc length = (150/360) $\times$ $2\pi$ $\times$ 12 = $10\pi$ cm.

• P1 Establishing $150/360\times 2\pi \times 12$ or an equivalent valid method.
• A1 Correct answer: $10\pi$

(a) $\frac{32}{3}\pi$

1. Multiply full-circle area by the fraction of a full turn.
   $$\frac{60}{360}\pi \times 8^{2}$$
2. Therefore $\frac{32}{3}\pi$.

• M1 Multiply full-circle area by the fraction of a full turn.
• A1 Correct answer: $\frac{32}{3}\pi$

(b) $\frac{8}{3}\pi +16$

1. Calculate the arc and add both straight radii.
   $$60/360\times 2\pi \times 8+2\times 8$$
2. Therefore $\frac{8}{3}\pi +16$.

• M1 Calculate the arc and add both straight radii.
• A1 Correct answer: $\frac{8}{3}\pi +16$

(a) $120$°

1. Compare the arc with a full circumference and multiply by 360.
   $$(6)/(2\times 9)\times 360$$
2. Therefore $120$°.

• M1 Compare the arc with a full circumference and multiply by 360.
• A1 Correct answer: $120$°

(b) $27\pi$

1. Use the same fraction of the full-circle area.
   $$\frac{120}{360}\pi \times 9^{2}$$
2. Therefore $27\pi$.

• M1 Use the same fraction of the full-circle area.
• A1 Correct answer: $27\pi$

(a) $8\pi$

1. 20 minutes is one third of a full turn.
   $$20/60\times 2\pi \times 12$$
2. Therefore $8\pi$.

• M1 20 minutes is one third of a full turn.
• A1 Correct answer: $8\pi$

(b) $48\pi$

1. Take one third of the area of the circle.
   $$\frac{20}{60}\pi \times 12^{2}$$
2. Therefore $48\pi$.

• M1 Take one third of the area of the circle.
• A1 Correct answer: $48\pi$

(a) 40.0 cm

1. Arc length.
   $$\tfrac{140}{360} \times 2\pi \times 9 = 7\pi = 21.99\ldots$$
2. The perimeter also includes two radii.
   $$7\pi + 9 + 9 = 39.99\ldots$$
3. Correct to 1 decimal place: 40.0 cm.

• P1 The arc length, $\frac{140}{360} \times 2\pi \times 9$.
• P1 Adding both radii to their arc length.
• A1 40.0 cm (awrt 40.0).

(a) $80$°

1. $$18\pi /(81\pi )$$
2. $$360\times 2/9$$
3. Sector area is (θ/360)$\pi r^{2}.$
4. $18\pi$ = (θ/360) $\times$ $81\pi .$
5. θ = 18 $\times$ 360 $\div$ 81 = $80^\circ.$

• P1 Establishing $18\pi /(81\pi )$ or an equivalent valid method.
• P1 Establishing $360\times 2/9$ or an equivalent valid method.
• A1 Correct answer: $80$°

(a) $150^\circ$

1. $$\tfrac{\theta}{360} \times \pi \times 6^2 = 15\pi$$
2. Divide by $36\pi$.
   $$\tfrac{\theta}{360} = \tfrac{15}{36} = \tfrac{5}{12} \;\Rightarrow\; \theta = 150$$

• P1 Setting up $\frac{\theta}{360} \times 36\pi = 15\pi$.
• A1 Correct answer: $150^\circ$.

(b) $12 + 5\pi$ cm

1. Arc length.
   $$\tfrac{150}{360} \times 2\pi \times 6 = 5\pi$$
2. Add the two radii: $12 + 5\pi$.

• M1 Arc length $\frac{150}{360} \times 12\pi$.
• A1 Correct answer: $12 + 5\pi$.

(c) 2.5 cm

1. The arc becomes the circumference of the cone's base.
2. $$2\pi R = 5\pi \;\Rightarrow\; R = 2.5$$

• P1 Recognising that the arc length equals the base circumference.
• A1 Correct answer: 2.5 cm.