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 Fig.). 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 π = 3.14)

We know that,  formula for the area of the sector of a circle

Area of the sector = $\frac{\theta }{{360}^0}$ $\times$ ${\mathrm{\pi r}}^2$

and

Area of square = (side)²

(i) From the figure we observe that the horse can graze an area of a sector of a circle with radius (r) 5m and an angle with a degree measure 90⁰ and it is a square filed 

Length of the rope = 5 m

Area of the field the horse can graze = Area of the sector with θ = 90⁰ and r = 5

= 90⁰/360⁰ $\times$ ${\mathrm{\pi r}}^2$

= 1/4 $\times$ π $\times$ (5 )²

= 25/4 $\times$ 3.14 

= 19.625 m²

(ii)  Now, we find the same for r = 10

Area of the field the horse can graze = Area of the sector with θ = 90⁰ and r = 10

= 90⁰/360⁰ $\times$ ${\mathrm{\pi r}}^2$

= 1/4 $\times$ 3.14 $\times$ 100 

= 78.5 m²

Thus, The increase in the grazing area

= 78.5  - 19.625 

= 58.875 m²