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.
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.
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 $\displaystyle 1/\sqrt{2}$
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.
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?
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.