Question 943435

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

if
{{{Volume= 126ft^3}}}
Height {{{h= x}}}
Length {{{L= x+3}}}
Width {{{W= x+4}}}

then

{{{126=(x+3)*(x+4)*x}}}

{{{126=(x^2+4x+3x+12)*x}}}

{{{126=x^3+7x^2+12x}}}

{{{0=x^3+7x^2+12x-126}}}........write {{{7x^2}}} as {{{-3x^2+10x^2}}} and {{{12x}}} as {{{-30x+42x}}}

{{{x^3-3x^2+10x^2-30x+42x-126=0}}}...group

{{{(x^3-3x^2)+(10x^2-30x)+(42x-126)=0}}}

{{{x^2(x-3)+10x(x-3)+42(x-3)=0}}}

{{{(x-3)(x^2+10x+42)=0}}}

real root is:

if {{{(x-3)=0}}} => {{{x=3}}}

here {{{(x^2+10x+42)=0}}} we have {{{a=1}}}, {{{b=10}}} and {{{c=42}}}

since {{{c>b}}}, determinant {{{b^2-4ac}}} is negative, so we will have two complex roots and we do not need them for length and width

so, we will use {{{x=3}}}

then

Height {{{h= 3ft}}}
Length {{{L= 6ft}}}
Width {{{W= 7ft}}}

check the volume:

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

{{{V=6ft*7ft*3ft}}}

{{{V=126ft^3}}} which is same as given