The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant? A. 1/2

B. 1

C. Sqrt 2

D. 2

E. 4

The answer to this question is D = 2

Explanation:

First of all, we know that the circumference of a circle C = 2πr

Thus, to get the rate at which the circumference of the circle is changing, we differentiate this above formula and we get

C' = 2π

Also, to calculate the area of a circle, we use the formula

A = πr^2

In similar manner, the rate at which the area is changing is to differentiate the formula for Area and thus we get;

A' = 2πr

From the question; At a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference.

Thus;

A' = 2C'

2πr = 2 (2π)

2πr = 4π

r = 4π/2π

r = 2

Since r = radius, the radius at that instant = 2