Question

Sides of a triangular fiald are 15 m, 16m and 17m. With the three cormers 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.

Answer 1

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)`