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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?

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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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