Question 1010402
{{{height=x}}}{{{in}}}
{{{width=x-1}}}{{{in}}}
{{{length=x+4}}}{{{in}}}
{{{system(Volume=height*width*length,Volume=12in^3)}}} ---> {{{x(x-1)(x+4)=12}}}
We need to solve the cubic equation {{{x(x-1)(x+4)=12}}} .
{{{x(x-1)(x+4)=12}}} <---> {{{x^3+3x^2-4x=12}}} <---> {{{x^3+3x^2-4x-12=0}}}
We can solve it by factoring, because the problem was designed to yield an integer {{{x}}} value.
We factor by parts:
{{{x^3+3x^2-4x-12=0}}}
{{{x^2(x+3)-4(x+3)=0}}}
{{{(x^2-4)(x+3)=0}}} ,
and then we factor further:
{{{(x-2)(x+2)(x+3)=0}}} .
The solutions are:
{{{x-2=0}}}<-->{{{x=2}}} ,
{{{x+2=0}}}<-->{{{x=-2}}} , and
{{{x+3=0}}}<-->{{{x=-3}}} .
We need a positive {{{x}}} , so {{{highlight(x=2)}}}--->{{{highlight(system(x-1=1,x+4=6))}}} .
So, the length, width, and height of the rectangular prism are
{{{highlight(system(length=6in,width=1in,height=2in))}}} .