Four cows are tethered at the four corners of a square field of side 50 m such that each can graze the maximum unshared area. What area will be left ungrazed? [Take π = 3.14.]
Here the 4 cows are tethered to each corner implies or means that the area available for grazing is a quadrant of radius 25 m with central angle as 60 degrees as the field is in shape of square . Now we need to find the area of this sector to find out the area available for grazing for all the cows and then subtract it from the total area of the square field to obtain the area left ungrazed.
The reason why we have taken the radius as 25 m is , basically we have considered that each cow is tethered to a rope which is equal to half of the side of the square as we had to maximize the area each cow gets to graze without sharing thus the maximum radius within which a cow can graze maximum unshared area is simply the half of the side of square.
Given the side of field which is in shape of square = a = 50 m
∴ Area of the field = Area of Square
⇒ Area of field = a2
⇒ Area of field = (502)
⇒ Area of field = 2500 m2
We know in an square all the angles are 90 degrees.
∴ Area available for grazing for one cow = area of sector/quadrant
Where R = radius of circle & θ = central angle of sector
Given R = 25 m & θ = 90°
Put the given values in the above equation,
⇒ Area available for grazing for one cow = 490.625 m2
⇒ Area available for 4 cows = 4 × Area available for one cow
⇒ Area available for 4 cows = 4 × 490.625
⇒ Area available for 4 cows = 1962.5 m2
Area left ungrazed = Area of field – Area available for grazing for 4 cows
⇒ Area that cannot be grazed = 2500 – 1962.5
⇒ Area that cannot be grazed = 537.5 m2
The area left ungrazed is 537.5 m2.