Sides of a triangular field are 15 m, 16m and 17m. With the three corners of the field a cow, a buffalo and a horse are tied separately with ropes of length 7m each to graze in the field. Find the area of the field which cannot be grazed by the three animals.

Explanation

Since with the three corners of the field a cow, a buffalo and a horse and tied separately with rope of length 7m each to graze in the field.
Area of field which cannot be grazed by animals = Area of `Delta BCA -` Area of three sectors
Here, `a = 15m, b = 16 m, c = 17 m`
`:. s = (a + b + c)/(2) = (15 + 16 + 17)/(2)`
`rArr s = (48)/(2) = 24 cm`
Area of `Delta BCA = sqrt(s (s - a) (s - b) (s - c))`
`= sqrt(24 (24 -15) (24-16) (24 - 17))`
`= sqrt(24 xx 9 xx 8 xx 7)`
`= sqrt(ul(2 xx 2) xx ul(2) xx 3 xx ul(3 xx 3) xx ul(2) xx ul(2 xx 2) xx 7)`
`rArr ar(DeltaBCA) = 24 sqrt(21) m^(2)`
Area of 3 secotrs `= (pi r^(2) theta_(1))/(360^(@)) + (pi r^(2) theta_(2))/(360^(@)) + (pi r^(2) theta_(3))/(360^(@))`
`= (pi r^(2))/(360^(@)) (theta_(1) + theta_(2) + theta_(3))`

`= (22)/(7) xx (7 xx 7)/(360^(@)) xx 180^(@) ( :. theta_(1) + theta_(2) + theta_(3) = 180^(@))`
`:.` Area of 3 sectors grazed by animals `= 77 m^(2)`
Hence, the area which cannot be grazed by 3 animals is equal to `(24 sqrt(21) - 77) m^(2)`