How quickly is her radius growing (in cm/day) when she can't fit through the circular stall door?

Question

Consider a spherical cow (!) that is consuming a GREAT deal of hay. She is fattening herself up at a rate of 200 liters/day. How quickly is her radius growing (in cm/day) when she can't fit through the circular stall door (which is 2 meters wide)?

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Answer 1 · 2018-02-21T21:27:57

#3.9\ \text{cm/day}#

#\frac{dr}{dt} = \frac{ \frac{dV}{dt} }{\frac{dV}{dr}}#

#{dV}/{dt} = 200\ \text{L/day} = 0.2 m^3\text{/day}#

#V = 4/3 pi r^3 #

#{dV}/{dr} = 4 pi r^2 = 16 pi##\ \ \ # (when #r = 2\ m#)

#{dr}/{dt} = {0.2\ m^3\text{/day}}/ (16 pi\ m^2)#

#= 0.1 / (8 pi) \text{m/day}#

# ~= 3.9\ \text{cm/day}#