Single Variable Calculus, Chapter 3, 3.8, Section 3.8, Problem 2

a.) Express $\displaystyle \frac{dA}{dt}$ in terms of $\displaystyle \frac{dr}{dt}$
b.) At what rate is the area of the spill increasing when the radius is 30m?

a.) Given: $A$, area of the circle

$\qquad r$, radius

Required: $\displaystyle \frac{dA}{dt}$ in terms of $\displaystyle \frac{dr}{dt}$

Solution: Let $A = \pi r^2$ be the area of circle where $r$ = radius

$\displaystyle \frac{dA}{dt} = \frac{dA}{dr} \left( \frac{dr}{dt} \right) = 2 \pi r \left( \frac{dr}{dt} \right)$

$\fbox{$\large \frac{dA}{dt} = 2 \pi r \left( \frac{dr}{dt} \right)$}$

b.) Given: $\displaystyle \frac{dr}{dt} = 1 m/s$

Required: $\displaystyle \frac{dA}{dt} = ?$ when $r = 30 m$

Solution: $\displaystyle \frac{dA}{dt} = 2 \pi (3)(1)$

$\fbox{$\large \frac{dA}{dt} = 60 \pi m^2/s$}$