SOLUTION: Q: find relative and exact dimensions of the circular cylinder of the largest lateral area which can be inscribed in a sphere of radius 8 inches [surface area of a cylinder, S = 2*

Question 221368: Q: find relative and exact dimensions of the circular cylinder of the largest lateral area which can be inscribed in a sphere of radius 8 inches [surface area of a cylinder, S = 2*pi*r*h. where r = radius, h = height.]
I think I can do this problem but i'm just confused as to what two equations to use and how and where to substitute. Please provide the work for the answer seperately so I have a chance to try it on my own but ill have your work just in case I need it anyway and to check my answer, but mostly I need a push in the right direction on how to set up and substitute the equations so I can solve them.
Thank You,
Bryan
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
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:

Now take a cross section of that to get

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

Since we have a triangle with legs of and along with a hypotenuse of , we can use the Pythagorean theorem to get the equation:

Start with the given equation.

Square to get

Multiply EVERY term by the LCD 4 to clear out the fraction.

Subtract from both sides.

Divide both sides by 4.

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

Now move onto the surface area of a cylinder formula.

Plug in

Plug in

Square 8 to get 64

Multiply

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 . Here are two ways to do this:

1) Derive 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 . 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.
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