Question

when the area in square units of an expanding circle is increasing twice as fast as its radius in linear units, the radius is

Explanation:

Step1: Write the area formula of a circle

The area formula of a circle is \(A = \pi r^{2}\).

Step2: Differentiate the area formula with respect to time \(t\)

Using the chain - rule, \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\).

Step3: Use the given relationship \(\frac{dA}{dt} = 2\frac{dr}{dt}\)

Substitute \(\frac{dA}{dt}=2\frac{dr}{dt}\) into \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\). We get \(2\frac{dr}{dt}=2\pi r\frac{dr}{dt}\).
Since \(\frac{dr}{dt}
eq0\) (the circle is expanding, so the radius is changing), we can divide both sides of the equation \(2\frac{dr}{dt}=2\pi r\frac{dr}{dt}\) by \(2\frac{dr}{dt}\).

Answer:

\(r=\frac{1}{\pi}\)