A balloon is being inflated at the rate of 3ft^3/min. How fast is the radius growing when the volume of the balloon is 100ft^3?

Question

amistre64 · Answer

looks like we need to work with a formula for the volume of a sphere.

anonymous · Answer

Would the usual volume rating work here? $$V= \frac{ 4 }{ 3 }\pi r ^{3}$$ Or will we need something different to do this?

amistre64 · Answer

thats a good start, now, the simplest way i know of doing this is to just take an implicit derivative.  what do we get?

anonymous · Answer

Something like? $$4\pi r^{2} $$

amistre64 · Answer

yes, but implicit assumes that V and r are functions of time so the chain rule applies

V' = 4pi r^2 r',  and we are looking for r'

r' = V'/(4pi r^2)

we know V' is 3,  all we need to do is determine r^2 when V=100

amistre64 · Answer

r = (3V/4pi)^(1/3)

r^2 = (3V/4pi)^(2/3)

r^2 = (300/4pi)^(2/3)

amistre64 · Answer

soo:$$r'=\frac{3}{4\pi r^2}$$
soo:$$\large r'=\frac{3}{4\pi (\frac{300}{4\pi})^{2/3}}$$