Question 221368
Let 


R = radius of the sphere
r = radius of the cylinder (ie the radius of the circular faces on the cylinder)
h = height of the cylinder



If we inscribe a cylinder in a sphere, we'll get the following:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step1-3.png">



Now take a cross section of that to get 


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step2-3.png">



Now draw in the diagonal of the rectangle along with an extra radius. Also, add the labels of 'h', 'R' and 'r' to their appropriate places


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step3-2.png">



Since we have a triangle with legs of {{{h/2}}} and {{{r}}} along with a hypotenuse of {{{R}}}, we can use the Pythagorean theorem to get the equation: {{{(h/2)^2+r^2=R^2}}}



{{{(h/2)^2+r^2=R^2}}} Start with the given equation.



{{{h^2/4+r^2=R^2}}} Square {{{h/2}}} to get {{{h^2/4}}}



{{{h^2+4r^2=4R^2}}} Multiply EVERY term by the LCD 4 to clear out the fraction.



{{{4r^2=4R^2-h^2}}} Subtract {{{h^2}}} from both sides.



{{{r^2=(4R^2-h^2)/4}}} Divide both sides by 4.



-----------------------------


{{{S=pi*r^2*h}}} Now move onto the surface area of a cylinder formula.



{{{S=pi*((4R^2-h^2)/4)*h}}} Plug in {{{r^2=(4R^2-h^2)/4}}}



{{{S=pi*((4*8^2-h^2)/4)*h}}} Plug in {{{R=8}}}



{{{S=pi*((4*64-h^2)/4)*h}}} Square 8 to get 64



{{{S=pi*((256-h^2)/4)*h}}} Multiply



{{{S=pi*(256h-h^3)/4}}} Distribute



Take note how the surface area is now a function of the height 'h'. In other words, the height solely determines the surface area of the cylinder.



The goal now is to maximize {{{S=pi*(256h-h^3)/4}}}. Here are two ways to do this:


1) Derive {{{S=pi*(256h-h^3)/4}}} with respect to 'h' and set that derivative equal to zero. Solve that equation to find the max.


2) Use a graphing calculator to find the highest point on {{{S=pi*(256h-h^3)/4}}}. The y-coordinate of this point will be the largest surface area while the x-coordinate will be the height.



I'll let you finish up.