# Determine the rate at which the air should be pumped into the balloon when its radius is 1mm. The best results will be achieved if the baloon's radius increases at a rate of 0.15mm/s. I need to...

Determine the rate at which the air should be pumped into the balloon when its radius is 1mm. The best results will be achieved if the baloon's radius increases at a rate of 0.15mm/s. I need to determine the rate at which the air should be pumped into the baloon when its radius is 1mm.

a) Assuming that the baloon is spherical.

b) Assuming that the balloon is cylindrical with length 1 cm.

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### 1 Answer

(a) We have a spherical balloon and we are asked to find the rate at which to pump air into the balloon when its radius is 1mm to achieve a rate of increase in the radius of .15mm/s

The volume of a sphere is `V=4/3pir^3` . We are looking for the rate of change of the volume with respect to time.

`(dV)/(dt)=4pir^2(dr)/(dt)` using the chain rule. We are given `r=1, (dr)/(dt)=.15` and we are looking for `(dV)/(dt)` . Substituting the known values we get:

`(dV)/(dt)=4pi(1)^2(.15)=.6pi~~1.885`

**So the rate to pump air into the balloon when the radius is 1mm is approximately 1.89 cubic mm per second.**

(b) If the balloon is cylindrical, we take the same approach. The volume of a cylinder is `V=pir^2h` . We are given the length h=1cm=10mm. Thus the volume is `V=10pir^2`

The change in volume is given by `(dV)/(dt)=20pir(dr)/(dt)` .

Substituting the known values we get:

`(dV)/(dt)=20pi(1)(.15)=3pi~~9.43`

**So the rate to pump air into the balloon when the radius is 1mm is approximately 9.43 cubic mm per second.**