Mathematics OpenStudy (anonymous):

Suppose a mothball loses volume at a rate proportional to its SURFACE AREA. If the radius of the ball decreases from 2cm to 1cm in 2 months, how long will it take till the radius is 1 mm? OpenStudy (dumbcow):

$\frac{dV}{dt} = k(4 \pi r^{2})$ $\frac{dV}{dt} = \frac{dV}{dr}*\frac{dr}{dt}$ $\frac{dV}{dr} = 4 \pi r^{2}$ $\rightarrow (4 \pi r^{2})\frac{dr}{dt} = k(4 \pi r^{2})$ $\frac{dr}{dt} = k \rightarrow dr =k*dt$ $\int\limits_{}^{}dr = \int\limits_{}^{} k dt$ $r = kt + C$ $r(0) = 2 \rightarrow C = 2$ $r(2) = 1$ $1 = 2k +2$ $\rightarrow k = -\frac{1}{2}$ $r = -\frac{1}{2}t +2 = \frac{1}{10}$ $t = \frac{19}{5} = 3.8 months$

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