Question

Air is pumped into a spherical balloon at a rate of 2 cm^3/s (two cubic centimetres per second). Denote by V the volume of the balloon, r the radius of the balloon, and S the surface area of the balloon. (a) Find dr/dt when V = 30 cm^3. (b) Find dS/dt when V = 30 cm^3. You may need these formulas: V = 4/3πr^3, S = 4πr^2.

Answer 1

V=4/3pi r^3
Find dV/dr from here

Answer 2

can you?

Answer 3

but its not dV/dt that i need to find, its dr/dt. Thats what i dont understand :(

Answer 4

ok... to start, you need to find r, when V = 30 cm^3

when you know that, use the relationship

\[\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}\]

you know DV/dt... and can find DV/dr when V = 30

once you have them, you can find dr/dt

Answer 5

ummm ok. So dies dV/dt = 2 then?

Answer 6

try to keep r as an exact value... makes it easier when substituting

Answer 7

thats correct... rate of change in volume with respect to time is

2 cm^3/sec

Answer 8

so where does the V=30cm^3 come in then?

Answer 9

you need to find the radius when the volume is 30cm...

then you substitute that into dV/dr

Answer 10

but i need to keep the answer in terms of dr/dt

Answer 11

yep.. and you need dr/dt when V = 30.... so you need to find the value of r when V = 30...


so solve

\[30 = \frac{4}{3} \pi r^3\]

for r...

this is the 1st step

Answer 12

so r would equal 1.9276? would i put it as a decimal?

Answer 13

ok... now find dV/dr

when

\[V = \frac{4}{3} \pi r^3\]

Answer 14

I would have left r as

\[r = \sqrt[3]{\frac{90}{4\pi}}\]

Answer 15

but a decimal will work fine

Answer 16

how do i find DV/dr?

Answer 17

do i subtitute the r value i found into the V formula?

Answer 18

differentiate the volume equation with respect to the radius...

Answer 19

do the derivative 1st...


then substitute

Answer 20

so V'=4(pi)r^2

and therefore dV/dr = 46.692 once i substitute r in

Answer 21

great... so then using the equation

\[\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt} \]

so

\[2 = 46.692 \times \frac{dr}{dt} \]

just solve for dr/dt

Answer 22

awesome thank you so much. so would i take the same steps for part b?

Answer 23

ok... so for the 2nd part...

you know

dr/dt

you need to find dS/dr.... using the surface area formula above....S = 4pi r^2

substitute your value of r.... you ahve calculated above then its a little easier

\[\frac{dS}{dt} = \frac{dS}{dr} \times \frac{dr}{dt}\]

hope this make sense

Answer 24

you are a legend, thank you so much. Heaps of help!!!!!

Answer 25

lol... hope its all correct... and good luck

Answer 26

do you mind checking part b. i got ds/dr=8(pi)(1.9276( which equals 48.446

so ds/dt = 48.446 x 0.0429 = 2.0783