Question

CALCULUS HELP?!!! Please?…
I really want to learn how to do this...
Gas is escaping from a spherical balloon at the rate of 10 cubic feet per hour. At what rate is the radius changing when the volume is 36 pi?

Answer 1

do you know the volume formula for a sphere?

Answer 2

yes @dpaInc
(4/3)pir^2

Answer 3

well, not quite... \(\large V=\frac{4}{3}\pi r^3 \)

taking the derivative with respect to time: \(\large V'=4\pi r^2\cdot r' \
before i explain further, do you understand how this was obtained?

Answer 4

\(\large V'=4\pi r^2\cdot r' \)
that's what it should read, sorry...

Answer 5

yessss! I just saw that . Yes yes . Its understandable you took the derivative of that and thats how you got v'=4pir^2(r')

Answer 6

ok..
V' represents the rate at which the volume is changing (given)
r' is the rate at which the radius is changing (what we're asked to find)

so \(\large r'=\frac{V'}{4\pi r^2} \), ok so far?

Answer 7

mmmmm…? So you basically divided it. to get?….

Answer 8

yes.. this is how we'll calclate r'... all we need is the values for V' and r...

now you need to find r when the volume of the sphere is 36pi cubic feet....
do you know how to do that?

Answer 9

noooo. I'll be honest. I don't know how to do that...

Answer 10

ok... we need to find the radius, r, when the volume, V, is 36pi so we'll use the volume formula:
\(\large V=\frac{4}{3} \pi r^3\)
\(\large 36\pi=\frac{4}{3} \pi r^3\) now we solve for r...
\(\large 36\cancel{\pi}=\frac{4}{3} \cancel{\pi} r^3\)
\(\large 36\cdot \frac{3}{4}= r^3\)
\(\large 27=r^3\)
\(\large 3=r\)

so radius is 3 when the volume is 36pi...

Answer 11

one last thing, do you remember what V' = ???

Answer 12

V' as in what it means?
V' is prime?

Answer 13

V' is the rate at which the volume is changing... it is given in the problem...

Answer 14

V' is the rate at which gas is escaping (volume changing) the sphere.

Answer 15

ok so how did 36pi take place of V?

Answer 16

Ohhhhh Ok nvm that question …. I got it.

Answer 17

since the gas is ESCAPING, this means the volume is changing -10 cubic feet per hour.
so V' = -10.
and we got r = 3 so now we'll put these values into the formula for r':
\(\huge r'=\frac{V'}{4\pi r^2}=\frac{-10}{4\pi (3)^2} \)

Answer 18

I was just about to ask what happens to the r'=v'/4pir^2

Answer 19

-71?

Answer 20

do you know what the units are for your answer?

Answer 21

or -70.69?

Answer 22

no...
it's best to leave the answer in the form above (in terms of pi)
if you want a decimal approximation then use your calculator....

Answer 23

oh ok… Yes the units would be ft.

Answer 24

ft per hour.

Answer 25

yes exactly.

Answer 26

thank you very muchhhhh! Im trying really hard to understand this.

Answer 27

also notice that it's a negative answer which means the radius is decreasing (which makes sense because gas is escaping and the volume is getting smaller)

Answer 28

yw...:)

Answer 29

do you mind helping me with one more? @dpaInc