A solid rectangular block has a square base .find its maximum volume if the sum of the height and any one side of the base is 12cm ?

Question
Answer 1

#256#

#V=a^2h# under the condition #h+a=12#

#V=V(a)=a^2*(12-a)=12a^2-a^3# Differentiating this formula with respect to #a# we obtain

#V'(a)=24a-3a^2# and again

#V''(a)=24-6a# #V'(a)=0# if #a=0# or #a=8# #a=0# is senseless, for #a=8# we get #V''(8)=24-48=-24<0# #V(0)=V(12)=0#

#V_max=64*4=256#

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