Question 581931
Let the length of the sides of the square base, in feet, be x.
The surface area of the base is {{{x^2}}} square feet.
The volume is the area of the base, {{{x^2}}}, times the height , h.
So,
{{{h*x^2=4}}} <--> {{{h=4/x^2}}}
The surface area of the sides can be calculated as perimeter of the base, {{{4x}}} times height, {{{4xh}}}, and substituting the expression for h found above:
lateral surface = {{{4x(4/x^2)=16/x)}}} (in square feet)
The cost for base and top, in $, is
{{{2*x^2*10=20x^2}}}
The cost for the sides is
{{{(16/x)*20=320/x}}}
Total cost is
{{{C(x)=20x^2+320/x}}}
{{{dC/dx=40x-320/x^2=(40x^3-320)/x^2}}}, which is zero for {{{x=2}}}
{{{C(2)=20*2^2+320/2=20*4+160=80+160=240}}}
The derivative is zero for {{{x=2}}}, negative for x<2, and positive for x>2, indicating a minimum of {{{C(x)}}} at {{{highlight(x=2)}}}, and that minimum cost is {{{highlight(C(2)=240)}}}