I need help on calculus! How do I use related rates? "A spherical balloon is inflated with helium at the rate of 100pi ft^3/min. How fast is the balloon's radius increasing at the instant the radius is 5 ft?"

Question

lgbasallote · Answer

the formula for volume of a sphere is $$\huge \frac 43 \pi r^3$$
right?

anonymous · Answer

Yup

lgbasallote · Answer

it says the balloon is inflated with the rate of 100 pi ft^3/min <--this is the rate the volume is increasing

in other words, it is the derivaitve of the volume

so...take the derivative of 4/3 pi r^3 and equate it to 100

got it?

anonymous · Answer

ohh thank you so much

lgbasallote · Answer

welcome

anonymous · Answer

that will give you the rate of increase of the radius

anonymous · Answer

but when do i use the 5 ft radius information

lgbasallote · Answer

so...do you know how to do the rest?

lgbasallote · Answer

take the derivative of 4/3 pi r^3 first

anonymous · Answer

so 4r^2pi

lgbasallote · Answer

close. it's actually $$\huge dV = 4\pi r^2 dr$$

lgbasallote · Answer

^this is where implicit differentiation is important

anonymous · Answer

the thing i don't udnerstand is where you got the dV and dr

lgbasallote · Answer

anyway... dr is the rate the radius is increasing

lgbasallote · Answer

dV is the derivative of volume

lgbasallote · Answer

let's try it this way...

$$\huge V = \frac 43 \pi r^3$$
$$\huge \frac{d}{dr} (V) = 4\pi r^2$$
agree?

anonymous · Answer

yes

anonymous · Answer

did you multiply each side by dr or something?

lgbasallote · Answer

now cross multiply... $$\huge dV = 4\pi r^2 dr$$

anonymous · Answer

yes i understand

lgbasallote · Answer

yes i did

lgbasallote · Answer

so anyway... you have $$\huge dV = 4\pi r^2 dr$$

you know the value of dV and the value of r...therefore, you can solve for dr

anonymous · Answer

OHH now i understand thanks :D

lgbasallote · Answer

welcome

anonymous · Answer

sorry it took a little long to catch on...i'm only a freshman

lgbasallote · Answer

no worries. i had difficulties with this one too

anonymous · Answer

thank you :)

lgbasallote · Answer

welcome again

anonymous · Answer

There is an intermediate step here that was sort of ignored.  You were never given $dV$ nor where you supposed to find $dr$.  You were given $dV/dt$ and asked for $dr/dt$.  So at some point, both sides should have been divided by $dt$.

lgbasallote · Answer

i'm really bad with proving that formula.....at least the answer stays teh same though.....