Question

A spherical balloon is being inflated at the rate of 8 cubic centimeters per second. At what rate is the radius changing when the volume = 10pi cc?

Answer 1

so if im not mistaken this is very simple problem -
normally \[\frac{ dv }{ dt }=4\pi r^2(\frac{ dr }{ dt })\]

Answer 2

Yes, pretty simple, so now just plug the numbers in :)

Answer 3

so if im trying to find out how fast the rate is changing then i need to move it around and solve for dr/dt
\[\frac{ dr }{ dt }=\frac{ \frac{ dv }{ dt } }{ V }\]
\[\frac{ dr }{ dt }=\frac{ 8 cc/\sec }{ 10 \pi cc } \]

Answer 4

so is am i correct in saying that the radius is changing at 2.513 / sec ?

Answer 5

if you dont mind all i did was just take 8/10pi and get 2.513?

Answer 6

V' = 4π r² r'
Now you need to find r² from V to plug in!

Answer 7

-> r' = V' / ( 4π r² )
= ...

Answer 8

arent i trying to find dr/dt? because i know that the V = 10pi ?

Answer 9

but the ( 4π r² ) doesnt that = 10π in this equation?

Answer 10

V' would = the 8 cc because that is the rate it is changing?

Answer 11

You have V, but you need r² from V !

Answer 12

super confused -
what i think i know >>
\[\frac{ dV }{ dt }=8 cc/\sec\]
\[V = 10\pi cc\]
what i need to know
\[\frac{ dr }{ dt }\]
\[\frac{ dr }{ dt } = \frac{ \frac{ dV }{ dt } }{ V }\]

Answer 13

V' = 4π r² r'
Do you agree up to this part?

Answer 14

yes

Answer 15

-> r' = V' / ( 4π r² )
Do you understand this?

Answer 16

yes i do :)

Answer 17

so in this equation i should be trying to find what r² is, correct?

Answer 18

Yes, sir,
what we have is V' = -8, but we still miss r² to fill in

Answer 19

since i dont know r' and i dont know r² i dont know how to solve the equation for r² :(

Answer 20

Using the original V = (4/3 ) π r³ to find r :
3* 10π/ 4π = r³

Answer 21

7.5π = r³ in this equation

Answer 22

r² = 5.732

Answer 23

r' = -8 / ( 4π (5.732 ) ?

Answer 24

3* 10π/ 4π = r³
-> r³ = 7.5

Answer 25

correct then r^2 would be 5.732

Answer 26

r' = V' / ( 4π r² ) so if i plug it back into the r' equation i should get
r' = -8 / ( 4π (5.732 ) correct?

Answer 27

No, r² = 3.832

Answer 28

sorry i feel dumb, pressing to many keys
r' = -8 / ( 4π (3.832)

Answer 29

So, you think you can handle from here :)

Answer 30

Thank you, sorry for being dumb :)

Answer 31

Take your time at start, then you'll be able to jump far in long run!