Show that the height of  cylinder of maximum volume that can be inscribed in a sphere of radius R is fraction numerator 2 straight R over denominator square root of 3 end fraction

 Also find maximum volume.

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

Expert Answer:

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

straight V equals πr squared straight h

Now from figure AB = h so that OB equals straight h over 2

            So we get straight r squared equals straight R squared minus open parentheses straight h over 2 close parentheses squared

        Using this the volume becomes      

V equals straight pi open parentheses straight R squared minus straight h squared over 4 close parentheses straight h
space space space equals πR squared straight h minus πh cubed over 4

Assuming  to be constant we differentiate with respect to  we get

            dV over dh equals πR squared minus fraction numerator 3 πh squared over denominator 4 end fraction

For maximum or minimum value

dV over dh equals 0
rightwards double arrow πR squared minus fraction numerator 3 πh squared over denominator 4 end fraction equals 0
rightwards double arrow straight h equals fraction numerator 2 straight R over denominator square root of 3 end fraction

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

            fraction numerator straight d squared straight V over denominator dh squared end fraction equals minus fraction numerator 3 πh over denominator 2 end fraction which of course is negative

So straight h equals fraction numerator 2 straight R over denominator square root of 3 end fraction is a point of maxima

Hence volume of cylinder is maximum when its height is fraction numerator 2 over denominator square root of 3 end fraction times the radius of outer sphere.

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