Find the dimension of the box that will minimize the total cost. Find the minimum total cost ?

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1 Answer
May 13, 2018

base length=height=#root(3)16000 # in
cost#= #Birr #26*root(3)16000#

Explanation:

My Pc
consider the block with square base of side length #a# and height #l#

for minimum cost it must have minimum surface area
however the volume is fixed #i.e# 16000 ci(=#V#, say)

we have volume #V=a*a*l=a^2l(=>l=V/a^2)#

we have surface area as function of side #a# and height #l# as

#S=2xx#(area of top (or bottom)) # + 4xx#(area of sides)#=2a^2+4al#
(this involves two variables(#a# and #l#,we will eliminate one of them))

substituting for #l#
#S=2a^2+(4aV)/a^2#
or #S=2a^2+(4V)/a#

for any function to be minimum or maximum its first derivative must be 0

we have
#(dS)/(da)=4a-(4V)/a^2#
for minimum surface area(or maximum) #(dS)/(da)=0#
#:.4a-(4V)/a^2=0=>4a=(4V)/a^2=>a^3=V=>a=V^(1/3)#

to check if it is point of maxima or minima we have second derivative test
we have
#(d^2S)/(da^2)=4+(8V)/a^3#
put #a=V^(1/3)#(the point where #(dS)/(da)=0#)
if it is positive it is point of minima if it is negative it is point of maxima(if it is 0 it is point of inflexion)

here
#((d^2S)/(da^2))_(x=V^(1/3))>0#
hence
#a=V^(1/3)# corresponds to point of minima
#:.a=root(3)16000# and #l=V/a^2=a^3/a^2=a#

#:.# dimensions are #base=height=root(3)16000 # in
area of bases=#2a^2=2*16000^(2/3)# sq. in
area of sides=#4al=4a^2=4*16000^(2/3)# sq.in
minimum cost= #9*2*16000^(1/3)+2*4*16000^(2/3)=Birr 26*16000^(2/3)#

by the way what currency is "Birr" : )