Question

@thesmartone

Answer 1

A spherical balloon filled with gas has a leak that causes the gas to escape at a rate of 1.5m^3/min. At what rate is the surface area of the balloon shrinking when the radius is 4m?

Would I use the same surface area formula from the last one, but solving for dS/dt?

Answer 2

That's correct

Answer 3

SA = \(4\pi r^2\)
dS/dt = \(4\pi (2r) \dfrac{dr}{dt}\)

r = 4m
dr/dt = 1.5m^3/min

Answer 4

Good try, but dr/dt is not 1.5 m^3/min. Pay attention to the units!

Answer 5

What would the 1.5 be then?

Answer 6

Is this one where I need volume and surface area?

Answer 7

What quantity is measured in m^3 ?

radius or volume ?

Answer 8

Volume

Answer 9

Ah, nice catch Ganeshie. I totally read over that. :)

Answer 10

That means 1.5 m^3/min is the rate of change of volume : dV/dt.
Not dr/dt

Answer 11

How would I use volume formula with this? Since it's asking for rate for surface area...

Answer 12

Would I use volume formula to find dr/dt?

Answer 13

Exactly !

V = 4/3 pi r^3

dV/dt = ?

(don't forget chain rule)

Answer 14

After simplifying, dV/dt = 4 pi r^2

Answer 15

dr/dt

Answer 16

Do you mean `dV/dt = 4 pi r^2 dr/dt ` ?

Answer 17

Yes, forgot to put dr/dt in the first comment haha

Answer 18

Looks good. Plug in dV/dt and you can solve dr/dt

Answer 19

Would r be 4 for this equation also?

1.5 m^3/min = 64\(\pi\)cm^2 dr/dt

\(\dfrac{1.5}{64\pi} m/min\) = dr/dt?

Answer 20

Looks good

Answer 21

So now I can use SA formula?

Answer 22

`1.5 = 4 pi r^2 dr/dt `

dr/dt = 1.5/(4pi r^2)

This is the expression for rate of change of radius. Plugging in r = 4 gives you the rate of change of radius exactly when r = 4.

Answer 23

Yes, you were stuck because you didn't know dr/dt. Now you have it..

Answer 24

dS/dt = \(64\pi(\dfrac{1.5}{64\pi})\)
64pi cancels leaving only 1.5?

Answer 25

1.5m^2/min?

Answer 26

Doesn't look correct..

S = 4pi r^2

dS/dt = 8pi r dr/dt

When r = 4,
dS/dt = 32pi * 1.5/(64pi)

Answer 27

Whoops, I was thinking r^2 instead of 2r, could the 32pi and 64 pi cancel leaving 1.5/2pi?

Answer 28

1.5/2

Answer 29

no piee

Answer 30

oh yeah :) m/min or m^2/min?

Answer 31

surface area is measured in m^2

its rate of change would be m^2/min

Answer 32

Thanks :) this stuff confuses the heck out of me.... trying to actually pass this test though haha

Answer 33

length measurements like radius, height, distance etc are measured in `meter`

area measurements are done in `m^2`

volume measurements are done in `m^3`

Answer 34

divide by `time` to get the corresponding units for rate of change

Answer 35

You're doing great... You might find it less confusing if you start taking units seriously :D