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?

Question

4823 views around the world

You can reuse this answer
Creative Commons License

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