Answer:
The polynomial function for the volume of the box ([tex]V[/tex]) in terms of [tex]x[/tex] is
[tex]V(x)= 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex].
Step-by-step explanation:
We present a representation of the specifications of the open box in a image attached below. The volume of the open box ([tex]V[/tex]), measured in cubic inches, is represented by this expression:
[tex]V = w\cdot h \cdot l[/tex]
Where:
[tex]w[/tex] - Width, measured in inches.
[tex]h[/tex] - Height, measured in inches.
[tex]l[/tex] - Length, measured in inches.
Polynomial functions in standard form are represented by the following form:
[tex]y = \Sigma_{i=0}^{n} c_{i}\cdot x^{i}[/tex]
Where:
[tex]n[/tex] - Order of the polynomial, dimensionless.
[tex]c_{i}[/tex] - i-th Coefficient, dimensionless.
[tex]x[/tex] - Indepedent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
If we get from figure that [tex]w = 12 - 2\cdot x[/tex], [tex]h = x[/tex] and [tex]l = 9 - 2\cdot x[/tex], then:
[tex]V = (12-2\cdot x) \cdot x \cdot (9-2\cdot x)[/tex]
[tex]V = (12\cdot x -2\cdot x^{2})\cdot (9-2\cdot x)[/tex]
[tex]V = 108\cdot x-18\cdot x^{2}-24\cdot x^{2}+4\cdot x^{3}[/tex]
[tex]V = 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex]
The polynomial function for the volume of the box ([tex]V[/tex]) in terms of [tex]x[/tex] is
[tex]V(x)= 108\cdot x -42\cdot x^{2}+4\cdot x^{3}[/tex].
