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January 25, 2015

Posted by **Hannah** on Friday, March 4, 2011 at 10:31pm.

Gas is escaping from a spherical balloon at the rate of 2ft^3/min. How fast is the surface area shrinking (ds/dt) when the radius is 12ft? (A sphere of radius r has volume v=4/3 pi r^3 and surface area S=4pi r^2.)

Remember that ds/dt = ds/dr X dr/dt

Step 1: Find ds/dr

I have no clue where to start. I was going to set 2 equal to 4 pi r^2 and then take the derivative but I really have no idea.

Step 2: Find dr/dr. (HInt dv/dt= dv/dr X dr/dt

Would I set 2 equal to volume and then take derivative?

Step 3: Find ds/dt?

Step 4: Evaluate ds/dt when the radius is 12ft.

After I find my equation I would just plug in 12 correct?

- Math(Please help) -
**agrin04**, Friday, March 4, 2011 at 11:46pmS = 4pi*r^2

dS/dr = 4pi*2r = 8pi*r

V = (4/3)pi*r^3

dV/dr = 4pi*r^2

dV/dt = dV/dr * dr/dt

2 = 4pi*r^2 * dr/dt

dr/dt = 1/(2pi*r^2)

dS/dt = dS/dr * dr/dt

= 8pi*r * 1/(2pi*r^2)

= 4/r

If r = 12, just plug the value to the last equation

- Math(Please help) -
**Hannah**, Saturday, March 5, 2011 at 12:52amHow did you get 12?

- Math(Please help) -
**Hannah**, Saturday, March 5, 2011 at 12:57amNevermind I meant how did you get 4/r but I figured it out. Thank You!!

- Math(Please help) -
**Hannah**, Saturday, March 5, 2011 at 1:53amIs this correct for the last part?

(8)(3.14)(12) X (1/(2)(3.14)(12^2) = 4/12

- Math(Please help) -
**agrin04**, Saturday, March 5, 2011 at 2:14amYes

- Math(Please help) -
**Hannah**, Saturday, March 5, 2011 at 2:23amOk do I actually have to do the calculations though?

- Math(Please help) -
**agrin04**, Saturday, March 5, 2011 at 6:40amIf you are not given the value, that means you have to write the steps I told you until the very last equation.

If you are given the value (of r), you don't have to put the value in every equation. Just follow my steps until the last equation, then plug the value after that. It's easier that way.

- Math(Please help) -
**Hannah**, Saturday, March 5, 2011 at 10:55amOK thank you!

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