Water is being pumped into a vertical cylinder of radius 5 meters and height 20 meters at a rate of 3 meters/min. How fast is the water level rising when the cylinder is half full?

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Answer 1 · 2017-02-20T18:14:01

I assume that there is an error and it should be #3# #m^3#/#min#

Variables
#V# = volume of water at time #t#
#h# = height of water at time #t#
(implicit variable: #t# = time in minutes)

(Note that the radius of the water is constant #5# #m#)

Rates of change

#(dV)/dt = 3# #m^3#/#min#
Find #(dh)/dt# when #h = 10# #" "# (When the height of the water is half the height of the cylinder.)

Equation relating the Variables

Volume of a cylinder: #V = pir^2h#

Volume of water:

#V = pi(5)^2h# or

#V = 25pih#

Equation relating rates of change

(Differentiate both sides of the last equation with respect to #t#.)

#d/dt(V) = d/dt(25pih)#

#(dV)/dt = 25pi(dh)/dt#

Finish

Substitute what we know, solve for what we want:

#3 = 25pi (dh)/dt#

#(dh)/dt = 3/(25pi)# #m#/#min#