Show that the height of cylinder of maximum volume that can be inscribed in a sphere of radius R is

Also find maximum volume.

### Asked by Topperlearning User | 19th Aug, 2014, 10:04: AM

In this question we see the radius of sphere is constant R. So we shall assume sphere is fixed. Also cylinder inside the sphere is variable. We need to find the dimensions of cylinder, which makes this of maximum volume.

Consider the cylinder of radius r and height h so its volume becomes

Now from figure AB = h so that

So we get

Using this the volume becomes

Assuming _{} to be constant we differentiate with respect to _{} we get

For maximum or minimum value

To check whether this is a point of local maxima or minima we differentiate once again with respect to h

which of course is negative

So is a point of maxima

Hence volume of cylinder is maximum when its height is times the radius of outer sphere.

### Answered by | 19th Aug, 2014, 12:04: PM

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