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

(a) if a is the area of a circle with radius r and the circle expands as time passes, find \\( \\frac { d a } { d t } \\) in terms of \\( \\frac { d r } { d t } \\).

(a) if a is the area of a circle with radius r and the circle expands as time passes, find \\( \\frac { d a } { d t } \\) in terms of \\( \\frac { d r } { d t } \\).

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 (A) with respect to (t) using the chain rule

By the chain rule (\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}). Differentiate (A = \pi r^{2}) with respect to (r): (\frac{dA}{dr}=2\pi r). So (\frac{dA}{dt}=2\pi r\cdot\frac{dr}{dt}).

Answer:

(2\pi r)