Mathematics

Problem

The radius

r

(cm) of a circular oil spill on water increases with time. When

r=10\,

cm, the measurements are

\dfrac{dr}{dt}=0.4\,\text{cm/s}

and

\dfrac{dA}{dt}=25\,\text{cm}^2\!/\text{s}

, where

A

is the area of the spill. a. (i) Using only the given measurements, compute the instantaneous rate of change of area with respect to radius,

\dfrac{dA}{dr}

, at this instant. b. (ii) Verify that your result is consistent with the geometrical relation between

A

and

r

for a circle. c. (iii) If subsequently

\dfrac{dr}{dt}

doubles while

\dfrac{dA}{dr}

remains the same as in part (i), determine the new rate at which the area is increasing.