What is the radius of an expanding circle when the area in square units of the circle is increasing four times as fast as its radius in linear units?

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The area of a circle can written as:

$$ A=\pi r^2$$

If we take the derivative of both sides with respect to time,

$$ \frac{dA}{dt}=\pi \frac{d}{dt}(r^2)$$ $$ \frac{dA}{dt}=\pi \cdot 2r\cdot \frac{dr}{dt}$$ $$ \frac{dA}{dt}=2\pi r\frac{dr}{dt}$$

The question mentions that the rate that the area is increasing is 4 times that of the radius:

$$ \frac{dA}{dt}=4\frac{dr}{dt} $$

Substituting this into the equation we found earlier,

$$ 4\frac{dr}{dt}=2\pi r\frac{dr}{dt}$$ $$ \boxed{\frac{2}{\pi}}=r$$