Q20 of 168 Page 231

Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

Let r be the radius and h be the height of the cylinder.

Then, the surface area (S) of the cylinder is given by:


S = 2πr2 + 2πrh


⇒ h



Let V be the volume of the cylinder. Then


V = πr2h



Now,


If


So, when then <0


Then, by second derivative test, the volume is the maximum when


Now, when . then h =


Therefore, the volume is the maximum when the height is the twice the radius or height is equal to diameter.


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