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 PcMy 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" : )