What is the technique of solving the following? Area of circle is increasing 3 times as fast as its radius in centimeters. What is the radius?

Question

I have a general idea that it is solved by using derivatives, but I am having a hard time converting text into equations?

Problem: Area of expanding circle is increasing 3 times as fast as its radius in centimeters. What is the radius?

Answer 1

"Area of expanding circle is increasing 3 times as fast as its radius" can be written mathematically as $$\frac{dA}{dt}=3\frac{dr}{dt}$$ because the derivative of the area is the rate at which the area changes and likewise for the radius. You also know that the area of a circle is $$A=\pi r^2.$$ Now take the derivative of the area formula so that you have a second formula for $\frac{dA}{dt}$. Can you see how to solve for the radius from here?

Answer 2

$\frac{d}{dr}A(r) = 3$ where $A(r) = \text{area of a circle with radius $r$}$.