Question

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.

Answer 1

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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.

cbse class12 bookproblem ch6 sec5 q20 p233sec-c medium math

  

 

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asked Nov 27, 2012 by thanvigandhi_1 
retagged Aug 16, 2013 by sharmaaparna1



1 Answer

Toolbox:Surface Area S=2πr2+2πrhS=2πr2+2πrhh=s−2πr22πrh=s−2πr22πrVolume V=πr2hV=πr2h

Step 1:

Let SS be the given surface area of the closed cylinder whose radius is rr and height hh.

Let VV be its volume.

Surface Area S=2πr2+2πrhS=2πr2+2πrh

h=S−2πr22πrh=S−2πr22πr----(1)

Volume V=πr2hV=πr2h

Substitute the value of h in the above equation

=πr2[S−2πr22πr]=πr2[S−2πr22πr]

=12=12r(S−2πr)r(S−2πr)

=12=12[Sr−2πr3][Sr−2πr3]

Differentiating with respect to r we get

dVdx=12dVdx=12[S−6πr2][S−6πr2]

Step 2:

For maxima and minima dVdxdVdx=0=0

S−6πr2=0S−6πr2=0

S=6πr2S=6πr2

From (1)

h=S−2πr22πrh=S−2πr22πr

Substitute the value os S in the above equation

h=6πr2−2πr22πrh=6πr2−2πr22πr

=4πr22πr=4πr22πr

=2r=2r

h=2rh=2r

Step 3:

On double differentiation of V we get,

d2Vdr2=12d2Vdr2=12(−12πr)(−12πr)

=−6πr=−6πr

=−ve=−ve

∴V∴V is maximum

Thus volume is maximum when h=2rh=2r

(i.e)when height of cylinder =diameter of the base.