Question 1116208
.
<pre>
I don't know about the cost of producing the box, and I think that this passage is not relevant to the rest of the condition.


But I know which dimensions will minimize the surface area of the box (which directly relate to the cost of the material).


The dimensions what minimize the surface area are 15 x 15 x 15 centimeters:  the box must be a cube.


It is easy to get this result analytically, using Calculus.


The surface area of the (x,y,z)-box is  A(x,y,z) = 2*(xy + xz + yz).


The volume = xyz = 3375,  so  z = {{{3375/xy)}}}.


Therefore, A(x,y,z) = {{{2*(xy + 3375/x + 3375/y)}}}  at the given volume.


The conditions  {{{(dA)/(dx)}}} = {{{(dA)/(dy)}}} = 0 give


    {{{x}}} - {{{3375/x^2}}} = 0  ====>  x^3 = 3375  ====>  x = {{{root(3,3375)}}} = 15,   and

    {{{y}}} - {{{3375/y^2}}} = 0  ====>  y^3 = 3375  ====>  y = {{{root(3,3375)}}} = 15.


And then  z = {{{3375/(xy)}}} = {{{3375/(15*15)}}} = 15.
</pre>

Solved.