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

Given: circle where its area is increasing at a uniform rate

To prove perimeter varies inversely as the radius

Explanation: Let the radius of the circle be ‘r’.

Let A be the area of the circle,

Then A = πr²……..(i)

As per the given criteria the area is increasing at a uniform rate, then

Now substituting the value from equation (i) in above equation, we get

Now differentiating with respect to t we get

Now let P be the perimeter of the circle, then

P = 2πr

Now differentiating perimeter with respect to t, we get

Applying the derivatives, we get

Now substituting value from equation (ii) in the above equation we get

Cancelling the like terms we get

Converting this to proportionality, we get

Hence the perimeter of the circle with given condition varies inversely as the radius.

Hence Proved