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e^(i Pi)+1=0
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Homework Statement
A snowball melts such that the volume decreases at a rate of 1cm3/min. At what rate is the diameter decreasing when diameter=10?
I know the answer is [itex]\frac{-1cm}{50∏ per ?}[/itex]. My problem is with the units on the bottom. It was given as seconds, but shouldn't it be per minute? There was no conversion done anywhere in the problem. The answer is small, and it seems like it would fit better as per seconds rather than per minute, but I can't find out how we went from minutes to seconds.
d=derivative
D=diameter
volume = [itex]\frac{4∏r^3}{3} = \frac{4∏(\frac{D}{2})^3}{3} = \frac{∏}{6}D^3[/itex]
[itex]\frac{dv}{dt}=\frac{-1cm^3}{min}=-1[/itex]
[itex]\frac{dv}{dt} = \frac{∏}{6}(3D^2)\frac{dD}{dt}[/itex]
[itex]-1 = \frac{∏}{6}(3D^2)\frac{dD}{dt}[/itex]
[itex]-1 = (\frac{D^2∏}{2})(\frac{dD}{dt})[/itex]
[itex]\frac{dD}{dt} = \frac{-1}{(D^2∏)/2}[/itex]
[itex]\frac{dD}{dt} = \frac{-2}{D^2∏}[/itex]
[itex]\left. \frac{dD}{dt}\right|_{D = 10} = \frac{-2}{(10)^2∏} = \frac{-2}{100∏} = \frac{-1}{50∏}[/itex]
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