If the area of circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius.

Explanation

Let the radius of circle= r And area of the circule, `A=pir^(2)`
`d/(dt)A= d/(dt) pir^(2)`
`(dA)/(dt) = 2pir. (dr)/(dt)`…………..(i)
Since, the area of a circule increases at a uniform rate. Then
`(dA)/(dt) =k` ……………….(ii)
where, k is a constant,
From Eqs. (i) and (ii), `2pir.(dr)/(dt) =k`
`rArr (dr)/(dt) = k/(2pir) = k/(2pi).(1/r)`..............(ii)
Let the permimeter, `P=2pir`
`(dP)/(dt) = d/(dt). 2pir rArr (dP)/(dt) = 2pi.(dr)/(dt)`
`=2pi. k/(pi) .1/2=k/r` [uisng Eq. (iii)]
`rArr (dP)/(dt) propto 1/r` Hence proved.