Suppose you are blowing a spherical bubble, filling it in with air at a uniform rate of 400mm^3/s. How fast is the radius of the bubble increasing by the time it is already 20mm long?

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Answer 1 · 2018-04-04T03:35:26

#1/(4pi)# #"mm/s"#

#V=4/3pir^3# (volume of sphere)

if #r=20mm#, #V=4/3pi(20mm)^3=32000/3pi(mm)^3#

differentiate both sides with respect to time:

#(dV)/dt=d/dt(4/3pir^3)#
#(dV)/dt=4pir^2*(dr)/dt#

volume of air increasing at uniform rate of 400mm^3/s = #(dV)/dt#

substitute: #400(mm)^3/s=4pir^2*(dr)/dt#

#(dr)/dt=400/(4pir^2)(mm)^3/s#
#(dr)/dt=400/(4pi(20mm)^2)(mm)^3/s# (#r=20mm#)
#(dr)/dt=1/(4pi)(mm)/s#

#(dr)/dt# is rate radius is increasing