A hemisphere is placed on a spherical bubble of radius 1. Find the maximum height of the bubble tower.

I'm a little confused on how to go about solving such a problem, would appreciate any advice you guys might have.

Thanks.

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- Sep 22nd 2008, 06:02 AMHweengee[SOLVED] Max/Min Word Problem
A hemisphere is placed on a spherical bubble of radius 1. Find the maximum height of the bubble tower.

I'm a little confused on how to go about solving such a problem, would appreciate any advice you guys might have.

Thanks. - Sep 22nd 2008, 01:48 PMshawsend
Alright, here's a plot. Now, suppose:

$\displaystyle \text{Green}=2\sqrt{\text{Red}^2-\text{Blue}^2}$. Can you then construct a function which relates the height of second hemisphere above the first (the orange line), as a function of the diameter of the second hemisphere (the green line)? Then take derivatives, set it equal to zero and get the maximum? I get $\displaystyle 1/\sqrt{2}$ - Sep 23rd 2008, 02:10 AMHweengee
the first figure is actually a sphere, but thanks for your help, i'll go work it out.

- Sep 23rd 2008, 10:18 AMHweengee
Is the value of 1/sqrt(2) which you got the radius of the small hemisphere? If so the maximum height would be just 1+ sqrt(2)?

- Sep 23rd 2008, 10:30 AMJameson
He knows that. It's just easier to look at it this way. You only need one part of the sphere at a time.

Like shawsend said, you need to find a way to express a function that outputs the height of the bubble over 1 (the orange line).

This is the hardest part in these problems. Making weird substitutions so you can get a function of one variable and maximize it. - Sep 23rd 2008, 11:03 AMshawsend
Hey, that triangle in there, isn't is just $\displaystyle g^2+b^2=r^2$ where $\displaystyle g$ is the radius of the small hemisphere (probably should have colored just half of the green line), b for blue, r for red?. Now, I want the blue line so that's $\displaystyle b=\sqrt{1-r^2}$. Now, look again at the figure. My ultimate goal is to express the height of the orange line as a function of the radius of the small hemisphere. Wouldn't that just be: $\displaystyle O(r)=r-1+\sqrt{1-r^2}$. Alright then. Now I have the height of the second hemisphere above the first as a function of the radius of the second. How do I find the maximum height?

- Sep 23rd 2008, 11:08 AMHweengee
i don't really get it, the height of the entire bubble tower would be blue line + r wouldn't it? so for an entire sphere it would be 1+r+sqrt(1-r^2)?

- Sep 23rd 2008, 11:14 AMshawsend
The entire height is blue+r. I'm just calculating the height of the bubble On the larger hemisphere (the orange line in the plot--click it to see a larger figure) and also I may have caused some confusion above. When I said I got $\displaystyle 1/\sqrt{2}$, I should have said that is the maximum radius of the small hemisphere. The orange height at that value is actually $\displaystyle O(1/\sqrt{2})=\sqrt{2}-1\approx=.414$ which you can read off of the plot.

I don't understand what you mean by the entire sphere. My understanding was take a sphere of radius 1 then put a hollow hemisphere on top of it. My drawing is a cross-section of half of the sphere and the hollow hemisphere. What then is the maximum height of the hollow hemisphere above the sphere which I calculated is about .414. - Sep 23rd 2008, 11:15 AMHweengee
yea i got it now, so the maximum height can be obtained by placing a hemisphere of radius 1/sqrt(2) on the sphere. thanks.