Optimization problem?

Question

A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of #32cm^3#. If nickel plating costs $1 per #cm^2# and silver plating costs $2 per #cm^2#, find the dimensions of the box to minimize the cost of the materials. (Round your answers to two decimal places.)

The box which minimizes the cost of materials has a square base of side length #?# cm and a height of #?# cm

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Answer 1 · 2018-03-09T10:44:48

#h=5.04cm# and #x=2.52cm#

let the side of the base #= x , cm # and the height #= h , cm#

So, if V is the volume then it is given by #V=x^2h# or #x^2h = 32#

but the Cost ($) , #C = 4x^2+4xh#

since we have two variable's then from #h=32/x^2#

#C=4x^2+4x(32/x^2)# so #C=4x^2+128/x#

then #C'=8x-128/x^2# but for minimum cost we put #C'=0#

then #128/x^2=8x# or # x^3=16or x=2.52cm# then from #h=32/x^2#

then #h=32/(16^(1/3))^2# then #h=5.04cm#

therefore the dimensions that minimize the cost are

#h=5.04cm# and #x=2.52cm#