A can of soda measures 2.5 inches in diameter and 5 inches in height. If a full-can of soda gets spilled at a rate of 4 in^3/sec, how is the level of soda changing at the moment when the can of soda is half-full?

1 Answer
Mar 31, 2015

The level is decreasing at a rate of #(4)/(1.56pi)"in/sec"#

The volume of liquid in the full can is given by:

#V=pir^(2)h=pi(d/2)^(2)h=(pid^(2))/(4).h# #color(red)((1))#

The rate of loss of liquid is given by:

#(dV)/(dt)=-4"in"^(3)"/s"#

We want to know how the level #h# is changing with time when #h=5#

From #color(red)((1))#:

#d=2.5#

So

#(dV)/(dh)=(6.25)/(4)pi=1.562pi# #color(red)((2))#

This is constant.

#(dV)/(dt)=(dV)/(dh)xx(dh)/(dt)#.

So from #color(red)((2))#:

#-4=1.562pixx(dh)/(dt)#

#(dh)/(dt)=-(4)/(1.562pi)#