Q. 234.8( 4 Votes )

Show that the rig

Answer :


Let the given volume be V and radius of base of cone be r and h be the height and l be the slant height of cone


Let S be the surface area


We have to find h such that the surface area is minimum so we have to establish a equation for S in terms of h


The surface area is given by S = πrl


Where l is slant height of cone given by



We need to eliminate r


For that given is volume of cone V




Square equation (a)


S2 = π2r2(r2 + h2)


Put value of r2





Now to find for what value of h S is minimum we first need to find the critical points such that


Differentiate S with respect to h and equate to 0




0 = -18V2 + 3Vh3π


0 = -6V + h3π


h3π = 6V




Now we have to check whether is point of minima or maxima


If then h is a point of minima


Let us check


From (i)



Differentiate again with respect to h


The LHS will be differentiated using the uv rule which states that (uv)’ = u’v + uv’ where u = 2S and




Substitute value of from (i)





Put






Now observe that S represents surface area V represents volume which are measures for some quantity hence both are positive π is a positive constant hence and hence is a point of minima


Now let us find h in terms of r by putting the value of volume



Since



h2 = 2r2


h = √2r


Hence for least curved surface area the altitude h has to be √2 times the base radius


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