Question 1078617
.
The solution by "josgarithmetic" is WRONG.
Below find the correct solution.
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<pre>
Bottom or top dimensions, 
length 3w, width w

h, for height


Given volume 4500,
{{{3w*w*h=4500}}}
{{{3w^2h=4500}}}
{{{h*w^2=1500}}}     (1)



Total surface area
S = {{{3w^2+3w^2+3wh+3wh+wh+wh}}} = {{{6w^2+8wh}}}    (2)


From (1), express h = {{{1500/w^2}}} and substitute it into (2). You will get

S = {{{highlight_green(6w^2+12000/w)}}}

-

  {{{(d/dw)(6w^2+12000/w)}}} = 
= {{{12w+12000(-1)w^(-2)}}}
= {{{(12w^3-12000)/w^2}}}
-

{{{d/dw = 0}}}

{{{12w^3-12000=0}}}
{{{w^3-1000=0}}}
{{{w^3=1000=10^3}}}
{{{highlight_green(w=10)}}}, width

-

Length, 3w
{{{highlight_green(30)}}}
</pre>

<U>Answer</U>.  Length = 30, width = 10, height = 15 inches.