A hemisphere of radius 7 sits on a horizontal plane. A cylinder stands with its axis vertical, the center of its base at the center of the sphere, and its top circular rim touching the hemisphere. Find the radius and height of the cylinder of maximum volume.
I have no clue how to go about solving this. Help, please?!?
Draw a right triangle inside the sphere, then we can see that
, where R is the radius of the sphere
and r and h are the radius and height of the cylinder, respectively.
The volume of the cylinder is
Set to 0 and solve for h. The rest will follow.
I tried what you said, and got 7(3^(1/2))/3 for h, which is incorrect, thereby making my value for r incorrect.
Originally Posted by galactus
What did I do incorrectly?
Well, solving dV/dh=0, we get
That appears correct to me.
You do realize this is the same thing as .
The denominator has been rationalized in your version, but they are equivalent.