$A$ horse is tied to a peg at one corner of a square-shaped grass field of side $15 \, m$ by means of a $5 \, m$ long rope (see figure). Find:
$(i)$ The area of that part of the field in which the horse can graze.
$(ii)$ The increase in the grazing area if the rope were $10 \, m$ long instead of $5 \, m$. (Use $\pi = 3.14$)

(N/A) From the figure,it can be observed that the horse can graze a sector of $90^{\circ}$ in a circle of $5 \, m$ radius.
Area that can be grazed by the horse $=$ Area of sector with radius $r = 5 \, m$ and angle $\theta = 90^{\circ}$.
Area $= \frac{\theta}{360^{\circ}} \times \pi r^{2}$
$= \frac{90^{\circ}}{360^{\circ}} \times 3.14 \times (5)^{2}$
$= \frac{1}{4} \times 3.14 \times 25$
$= 19.625 \, m^{2}$
Area that can be grazed by the horse when the length of the rope is $10 \, m$ long $(r = 10 \, m)$:
Area $= \frac{90^{\circ}}{360^{\circ}} \times \pi \times (10)^{2}$
$= \frac{1}{4} \times 3.14 \times 100$
$= 78.5 \, m^{2}$
Increase in grazing area $= (78.5 - 19.625) \, m^{2}$
$= 58.875 \, m^{2}$