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# 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 $\pi = 3.14$)

"

Given:

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.

To do:

We have to 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.

Solution:

(i) As given in the question,

The maximum area can be grazed by the horse is the area of the quadrant of the circle with radius $r=5\ m$.

Therefore,

Area grazed by the horse$=$ Area of the quadrant $( 5\ m)$

$=\frac{1}{4}\pi r^2$

$=\frac{1}{4}\times3.14\times5^2$

$=\frac{78.50}{4}$

$=19.625\ m^2$

Therefore, $19.625\ m^2$ is the area of the field in which the horse can graze.

(ii) Length of the rope is increased from $5\ m$ to $10\ m$

This implies,

New radius of the sector grazed by the horse $= 10\ m$

Therefore,

Area grazed by the horse$=$ Area of the quadrant $( 10\ m)$

$=\frac{1}{4}\pi r^2$

$=\frac{1}{4}\times3.14\times(10)^2$

$=25\times3.14$

$=78.5\ m^2$

This implies,

Increase in the grazing area $=78.5-19.625$

$=58.875\ m^2$

Therefore, increase in the grazing area is $58.875\ m^2$.

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