calculus
posted by Debbie .
for a given patient it is determined that the blood vessel may rupture if its diameter increases faster than 200 um/s. The blood vessel starts with diameter 4mm and needs to expand to a diameter of 7mm. Is it safe to inflate the balloon at a constant rate of 30 mm^3/s? What about 60 mm^3/s.?

From the previous post, keeping all the measurements in mm,
dV/dt = (4πr^2 + 60π) dr/dt
so, if dV/dt = 30,
30 = 4π(16 + 15) dr/dt
so
dr/dt = .0770 mm/s = 77 um/s. That means that the diameter is increasing by 154um/s, so that's ok. As the diameter increases, the rate of expansion slows down.
At 60mm^3/s, the inflation rate starts at 308um/s. Boom!