If the area of an expanding circular region increases at a constant rate with respect to time, then the rate of increase of the perimeter with respect to the time

Options

A. Varies inversely as radius
B. Varies directly as radius
C. Remains constant
D. Varies directly as square of the radius

Answer

A

Explanation

Let A be the area and P be the perimeter of the circular region of radius r. Then,
`A = pir^(2) and P=2 pir`
`implies (dA)/(dt)=2 pir (dr)/(dt) and (dP)/(dt)=2 pi (dr)/(dt)`
It is given that `(dA)/(dt)=k` (constant).
`implies (dr)/(dt)=(k)/(2pir)`
`therefore (dP)/(dt)=2pixx(k)/(2pir)=(k)/(r)prop(1)/(r)`