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(C) The horses are tethered at the vertices of the triangle with rope length $r = 7 \, m$. The area grazed by each horse is a sector of a circle with

Vedclass · Accepted Answer

$77$

Vedclass · Answer

(C) The horses are tethered at the vertices of the triangle with rope length $r = 7 \, m$. The area grazed by each horse is a sector of a circle with radius $r$ and central angle equal to the interior angle of the triangle at that vertex.Let the angles at vertices $A, B,$ and $C$ be $\angle A, \angle B,$ and $\angle C$ respectively.The sum of the interior angles of a triangle is $180^{\circ}$.Total grazed area $= 	ext{Area of sector at } A + 	ext{Area of sector at } B + 	ext{Area of sector at } C$$= \frac{\angle A}{360^{\circ}} \pi r^2 + \frac{\angle B}{360^{\circ}} \pi r^2 + \frac{\angle C}{360^{\circ}} \pi r^2$$= \frac{(\angle A + \angle B + \angle C)}{360^{\circ}} \pi r^2$Since $\angle A + \angle B + \angle C = 180^{\circ}$,Total grazed area $= \frac{180^{\circ}}{360^{\circ}} \pi r^2 = \frac{1}{2} \pi r^2$$= \frac{1}{2} 	imes \frac{22}{7} 	imes 7^2 = \frac{1}{2} 	imes \frac{22}{7} 	imes 49 = 11 	imes 7 = 77 \, m^2$.

$3$ horses are tethered at $3$ corners of a triangular plot of land having sides $20 \, m$,$30 \, m$,and $40 \, m$,each with a rope of length $7 \, m$. The area (in $m^2$) of the region of this plot,which can be grazed by the horses,is (Use $\pi = \frac{22}{7}$)

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