Hello, totalnewbie!
Cylinder is inscribed in a sphere of radius $\displaystyle R$.
What is the biggest volume of cylinder then? Code:
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The radius of the cylinder is $\displaystyle r$.
The height of the cylinder is $\displaystyle 2h$.
The volume of the cylinder is: .$\displaystyle V \:=\:\pi r^2(2h) \:=\:2\pi r^2h$ [1]
In the diagram, we see that: .$\displaystyle r^2 + h^2\:=\:R^2\quad\Rightarrow\quad h \:=\:\sqrt{R^2 - r^2}$ [2]
Substitute [2] into [1]: .$\displaystyle V \:=\:2\pi r^2\sqrt{R^2-r^2}$
Therefore, we must maximize: .$\displaystyle V \;= \;2\pi r^2\left(R^2-r^2\right)^{\frac{1}{2}}$
. . Go for it!