Question

a lawn is in a field in the shape of a triangle, abc.
ac = 21 metres, ab = 23 metres and ( angle cab = 70^circ )
a goat is in an adjacent field in the shape of a sector of a circle with centre, a, and radius 15 metres.
the fields are shown in the diagram.
diagram not to scale
determine which animal, the sheep or the goat, is in the field with the larger area, and state how many square metres larger this area is.

Explanation:

Step1: Calculate sheep pen area

Use triangle area formula: $A=\frac{1}{2}ab\sin\theta$
$A_{sheep} = \frac{1}{2} \times 21 \times 15 \times \sin(74^\circ)$
$\sin(74^\circ) \approx 0.9613$
$A_{sheep} \approx \frac{1}{2} \times 21 \times 15 \times 0.9613 \approx 151.4$

Step2: Calculate goat pen area

Use sector area formula: $A=\frac{\theta}{360^\circ} \times \pi r^2$
The sector angle is $360^\circ - 74^\circ = 286^\circ$, $r=15$
$A_{goat} = \frac{286}{360} \times \pi \times 15^2$
$A_{goat} = \frac{286}{360} \times \pi \times 225 \approx 0.7944 \times 706.86 \approx 561.5$

Step3: Compare and find difference

$561.5 > 151.4$, so goat pen is larger.
Difference: $561.5 - 151.4 = 410.1$

Answer:

The goat's field has the larger area. It is approximately 410 square metres larger than the sheep's field.