Question

find the circular cylinder of largest lateral area which can be inscribed in a sphere of radius R=4 feet

Answer 1

Let say radius is r 
its height is h 
its lateral area = y 
y = 2 pi r h 
since the cylinder is inscribed in the sphere 
So (2r )^2 + h^2 = 64 
then 4 (r^2) = 64 - h^2 
since y^2 = 4 (pi)^2 r^2 h^2 
then y^2 = (pi)^2 *h^2 * (64 -h^2) 
y^2 = 64 (pi)^2 * h^2 - (pi)^2 * h^4 
2 y y' = 128 (pi)^2 * h - 4 (pi)^2 * h^3 
putting y' = 0 
4 (pi)^2 h ( 32 - h^2)=0 
ether h = 0 testing this value (changing of the sign of y' before and after ) y is minimum 
or h = 4 sqrt(2) 
testing this value (changing of the sign of y' before and after ) y is maximum 
So the maximum value of y^2 = (pi)^2 *32 *( 64 - 32) 
y^2 = (pi)^2 * (32)^2 
y = 32 (pi) square feet

hope this helps