Answer:
The values of x for which the model makes sense is [tex]x\ge 6\,ft[/tex].
Step-by-step explanation:
From statement we have the following expression for the volume of the rectangular room is:
[tex]V(x) = 2\cdot x^{3}-14\cdot x^{2}+12\cdot x[/tex] (1)
Where:
[tex]V[/tex] - Volume of the rectangular room, measured in cubic feet.
[tex]x[/tex] - Length of the room, measured in feet.
By Algebraic means we factorize the polynomial:
[tex]V(x) = x\cdot (2\cdot x^{2}-14\cdot x +12)[/tex]
[tex]V(x) = x\cdot (x-6)\cdot (x-1)[/tex]
Given that both volume and length must be positive variables, then the following conditions must be satisfied:
[tex]x \ge 0\,ft\,\land\,x\ge 6\,ft\,\land\,x\ge 1\,ft[/tex]
Then, the values of x for which the model makes sense is [tex]x\ge 6\,ft[/tex].
