A person observed that he required 30 seconds less time to cross a circular ground along its diameter than to cover it once along the bound…

Let r be the radius of the circular ground in meters

Let V be the speed of the man in m/s

V = 30 m/minute = (30/60) m/s = 0.5 m/s

Let ‘t’ be the time required to cover the field along the boundary by the man in seconds

So, length of the path = (perimeter of the circle)/2 = 2πr/2

(πr)/V = t

Since it requires 30 seconds less to cover the field while going through the diameter

So,

(2r)/V = t – 30

Putting the values of t and V

⇒ 2r = 0.5 ×[{(πr)/0.5} – 30]

⇒ 4r = 2πr – 30

⇒ 2r(2-π) = -30

⇒ r = 15/(π-2)

⇒ r = 13.13 m.