Sheep’s field (triangle ABC): Use area formula \( \text{Area} = \frac{1}{2} \times AB \times AC \times \sin(\angle A) \) (M1)
Substitute: \( \frac{1}{2} \times 15 \times 21 \times \sin(78^\circ) \approx \frac{1}{2} \times 15 \times 21 \times 0.978147 \approx 154.058 \, \text{m}^2 \) (A1)
Goat’s field (sector): Use area formula \( \text{Area} = \frac{\theta}{360^\circ} \times \pi \times r^2 \) (M1)
Substitute: \( \frac{78^\circ}{360^\circ} \times \pi \times 8^2 = \frac{78}{360} \times \pi \times 64 \approx 0.216667 \times \pi \times 64 \approx 157.498 \, \text{m}^2 \) (A1)(A1)
Alternative for goat’s field: Total circle area \( \pi \times 8^2 = 64\pi \), sector area \( \frac{78}{360} \times 64\pi \approx 157.498 \, \text{m}^2 \) (M1)(A1)
Compare: \( 157.498 > 154.058 \), goat’s field is larger (A1)
Difference: \( 157.498 – 154.058 \approx 3.44 \, \text{m}^2 \) (A1)
Result: Goat’s field is larger by 3.44 m² (3.44026…) [6]