Question

(a) if ( a ) is the area of a circle with radius ( r ) and the circle expands as time passes, find ( \frac{da}{dt} ) in terms of ( \frac{dr}{dt} ).
( \frac{da}{dt}=(quad) \frac{dr}{dt} )

Explanation:

Step1: Write the formula for the area of a circle

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

Step2: Differentiate both sides with respect to \(t\)

Using the chain - rule \(\frac{dA}{dt}=\frac{d}{dt}(\pi r^{2})\). By the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and chain - rule \(\frac{d}{dt}(u^{n})=nu^{n-1}\frac{du}{dt}\) (where \(u = r\) and \(n = 2\)), we have \(\frac{dA}{dt}=\pi\times2r\times\frac{dr}{dt}\).

Answer:

\(2\pi r\)