A horse is tethered to one corner of a field which is in the shape of an equilateral triangle of side 12 m. If the length of the rope is 7 m, find the area of the field which the horse cannot graze. Take √3 = 1.732. Write the answer correct to 2 places of decimal.
Here the horse is tethered to one corner implies or means that the area available for grazing is a sector of radius 21 m with central angle as 60 degrees as the field is in shape of equilateral triangle . Now we need to find the area of this sector to find out the area available for grazing and then subtract it from the total area of the triangular field to obtain the area left ungrazed.
Given the side of field = a = 12 m
∴ Area of field = Area of equilateral triangle
⇒ Area of field = 62.352 m2
We know in an equilateral triangle all the angles are 60 degrees.
∴ Area available for grazing = Area of the sector
Where R = radius of circle and θ = central angle of sector
Given R = 7 m and θ = 60°
Put the given values in the above equation,
⇒ Area available for grazing = 25.666 m2
Area that cannot be grazed = Area of field – Area available for grazing
⇒ Area that cannot be grazed = 62.352 – 25.666
⇒ Area that cannot be grazed = 36.686 m2
The area that cannot be grazed is 36.656 m2.