Question 800671
{{{V =L*W*h }}}

given:

{{{V =256in^3}}}
{{{L=2W}}}

so, you have

{{{256in^3 =2W*W*h }}}

{{{256in^3 =2W^2*h }}}

{{{256in^3/2W^2=h }}}

{{{128in^3/W^2=h }}}......height


{{{S=2LW+2WH+2LH}}}


{{{352in^2=2*2W*W+2W(128in^3/W^2)+2*2W(128in^3/W^2)}}}

{{{352in^2=4W^2+2cross(W)(128in^3/W^cross(2))+4cross(W)(128in^3/W^cross(2))}}}

{{{352in^2=4W^2+2(128in^3/W)+4(128in^3/W)}}}

{{{352in^2=4W^2+256in^3/Win+512in^3/Win}}}

{{{352in^2*W=4W^3+256in^2+512in^2}}}

{{{352in^2*W=4W^2+768in^2}}}

{{{4W^3-352in^2*W+768in^2=0}}}

{{{W^3-88in^2*W+192in^2=0}}}

{{{(W-8)(W^2+8W-24) = 0}}}



if {{{(W-8)  = 0}}} => {{{W=8in}}}.....one solution

if {{{W^2+8W-24 = 0}}} => {{{W = (-8 +- sqrt( 8^2-4*1*(-24)))/(2*1) }}}

{{{W = (-8 +- sqrt( 64+96))/2 }}}

{{{W = (-8 +- sqrt( 160))/2 }}}

{{{W = (-8 +- 12.65)/2 }}}

other solution:

{{{W = (-8 + 12.65)/2 }}}.....we need only positive value since the width cannot be negative value

{{{W =4.65/2 }}}

{{{W =2.325in }}}




The width of the box in inches can be: {{{W=8in}}} or {{{W =2.325in }}}

if {{{W=8in}}} => {{{L=16in}}} and {{{128in^3/64in^2=h }}}=>{{{2in=h }}}

if {{{W =2.325in }}}=> {{{L=4.65in}}}and {{{128in^3/5.4in^2=h }}}=>{{{23.68in=h }}}


{{{ graph( 600, 600, -20, 20, -40, 10, (x-8)(x^2+8x-24)) }}}