Question 695021
{{{540=h^3+7h^2+12h}}} --> {{{h^3+7h^2+12h-540=0}}}
Any rational zero of the polynomial {{{P(h)=h^3+7h^2+12h-540}}} will be a factor of {{{540}}}, with a + or a - sign.
{{{540=2^2*3^3*5}}}
You know we are looking for a positive solution, so I would try + signs first.
The choices for factors are 1,2,3,4,5,6,9,10,12,15,18,20,27,30,36,45,54,60,90,108,135,180,270, and 540.
It is obvious that for positive {{{h}}} values, as {{{h}}} increases, {{{P(h)}}} increases.
{{{P(1)=1+7+12-540<0}}} and {{{P(10)=1000+700+120-540>0}}},
so the solution could be one of the factors in between 1 and 10: 2,3,4,5,6,and 9.
{{{P(5)=125+7*25+12*5-540=125+175+60-540=360-540=-180}}},
so we expect that the solution could be one of the factors between 5 and 10 (either 6 or 9).
{{{P(6)=216+7*36+12*6-540=216+252+72-540=0}}}, so {{{highlight(h=6)}}} is one solution.
Is there another solution?
Of course not.
If we make {{{h}}} smaller, the width and length will be smaller, and the volume will have to be smaller.
If we make {{{h}}} larger, the width and length will be larger, and the volume will have to be larger.