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

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)?

1 Answer
Feb 21, 2018

3.9\ \text{cm/day}

Explanation:

\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}