If the area of a circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius.
Given: A circle with uniformly increasing area rate
To prove: relation between perimeter and radius is inversely proportional
Explanation: Take the radius of circle ‘r’
Let the area of the circle be A
Then ……..(i)
After considering the give criteria of area increasing at a uniform rate,
Substitute the value of equation (i) into above equation,
Differentiating with respect to t results in
Let P as the perimeter of the circle,
P = 2πr
Now differentiate the perimeter with respect to t,
Apply all the derivatives,
Substituting of equation (ii) in the above equation,
Cancelling out of the like terms results into
Now covert it into proportionality,
Hence in the given conditions, the relation between perimeter of circle and radius is inversely proportional.
Hence Proved