Question

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

Answer 1

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

Explanation:

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" : )