



 
Hi Jim Using the information you gave me, I came up with the following diagrams
The volume of the cylinder is V=πx^{2}h. The radius of the sphere and the radius of the cylinder's base form a triangle with the height of the cylinder. Using the Pythagorean Theorem we can say x^{2}+(½ h)^{2}=r^{2} Using implicit differentiation with respect to x, we arrive at the following
Since the radius is not changing dr/dx = 0 2x+½ h dh/dx = 0 You can now solve for dh/dx from the above equation and substitute it into the change of volume formula dh/dx = 4x/h dV/dx = 2πxh + πx^{2}dh/dx= 2πxh + πx^{2}(4x/h) At minimum or maximums, rates of change equal zero. So for the maximum volume dV/dx=0 0 = 2πxh + πx^{2}dh/dx= 2πxh + πx^{2}(4x/h) You can now solve for x in terms of h then use the Pythagorean Theorem to find the dimensions of your maximum cylinder. Hope this helps, Janice
 


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