



 
The volume of a cylinder is
where r is its radius and h is its height. You need to differentiate this expression and then use what you know about calculus to find the values of r and h that maximize the volume. The challenge is that V is a function of two variables, r and h and you need to express V as a function of one variable to differentiate it. You need to use that fact that the cylinder is inside a sphere to eliminate one variable.
In the diagram C is the centre of the sphere and triangle ABC is a right triangle. Use Pythagoras' theorem to find a relationship between r^{2} and h^{2}. Use this relationship to eliminate one variable in equation (1) and then use calculus to maximize the volume of the cylinder. I hope this helps,  


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