(a) if ( a ) is the area of a circle with radius ( r ) and the circle expands as time passes, find (…

(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} ).\n( \frac{da}{dt}=(quad) \frac{dr}{dt} )

(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} ).\n( \frac{da}{dt}=(quad) \frac{dr}{dt} )

Answer

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)