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Water draining out of spherical bowl

Hi,

Its been 3 years since I've posted here. And I find myself trying to review the basics of calc before moving on to bigger things.

I found this problem in my old math book:

Attachment 29083

Sorry I couldn't rotate it I'm in a rush I tried to look online for some online image utility... oh well.

So, water is flowing out of a half hemisphere blown (the bottom half, lol) at a constant rate of 6 m^3/min. Radius of sphere is R (13 meters), radius of water level is r. height of water is y Volume is V. formula of Volume at height y is:

Find an equation for r and find dr/dy at y=8 meters

I assume, that , baseless assumption I admit, so that using chain rule:

But that doesn't include the volume in any way. If correct it gives me a relationship between r and y (radius of water level vs depth)

I'm also curious if I can derive an actual formula for r with what is given.

This isn't homework I'm an adult returning to higher education and I'm doing a crash course math review on my own before some harder math classes the next semesters coming up. My algebra needs work.

Re: Water draining out of spherical bowl

I don't see how you get - how did you get that? From Pythagoras its apparent that , so .

Question C asks for the rate of change of r, or dr/dt, not dr/dy. You can get that using the chain rule:

Re: Water draining out of spherical bowl

Wow that looks obvious. Thank you.