Applications of the derivative (max and min problem)

[electmenot]

I'm lost on how to set this problem up:

Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 3 inches.

I have:

V=pi r^2 h
A= 2pi r h + 2pi r ^2

But beyond that I am lost, any help would be greatly appreciated.

Kris

 

[stapel]

Since you're working with volumes, I'm not sure why you're using the surface-area formula...?

Draw the sphere as a circle. Draw the cylinder inside the sphere as a rectangle with the four corners touching the circle.

From the center, draw an horizontal line sideways (say, to the right side of the rectangle) to indicate the radius of the cylinder. Label this as "r".

From the center, draw a vertical line (say, to the top of the rectangle) to indicate half of the height of the cylinder. Label this as "h/2".

From the center, draw a diagonal-upward line to the top right corner where the rectangle meets the circle. By definition, this is a radius line of the sphere, so label this line as "3".

Note that these three lines, together with the outline of the rectangle, form two right triangles, each having legs of lengths h/2 and 4, and each having an hypotenuse of length 3. Use the Pythagorean Theorem to solve for h in terms of r.

Use the volume-of-a-cylinder equation, and plug in for h in terms of r. Then maximize.

If you get stuck, please reply showing how far you have gotten. Thank you.

Eliz.