# 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 $\pi = 3.14$)"

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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 $58.875\ m^2$.

Updated on 10-Oct-2022 13:23:55