Answer:
The polynomial function for the volume of the box (
) in terms of
is
.
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 (
), measured in cubic inches, is represented by this expression:
![V = w\cdot h \cdot l](https://tex.z-dn.net/?f=V%20%3D%20w%5Ccdot%20h%20%5Ccdot%20l)
Where:
- Width, measured in inches.
- Height, measured in inches.
- Length, measured in inches.
Polynomial functions in standard form are represented by the following form:
![y = \Sigma_{i=0}^{n} c_{i}\cdot x^{i}](https://tex.z-dn.net/?f=y%20%3D%20%5CSigma_%7Bi%3D0%7D%5E%7Bn%7D%20c_%7Bi%7D%5Ccdot%20x%5E%7Bi%7D)
Where:
- Order of the polynomial, dimensionless.
- i-th Coefficient, dimensionless.
- Indepedent variable, dimensionless.
- Dependent variable, dimensionless.
If we get from figure that
,
and
, then:
![V = (12-2\cdot x) \cdot x \cdot (9-2\cdot x)](https://tex.z-dn.net/?f=V%20%3D%20%2812-2%5Ccdot%20x%29%20%5Ccdot%20x%20%5Ccdot%20%289-2%5Ccdot%20x%29)
![V = (12\cdot x -2\cdot x^{2})\cdot (9-2\cdot x)](https://tex.z-dn.net/?f=V%20%3D%20%2812%5Ccdot%20x%20-2%5Ccdot%20x%5E%7B2%7D%29%5Ccdot%20%289-2%5Ccdot%20x%29)
![V = 108\cdot x-18\cdot x^{2}-24\cdot x^{2}+4\cdot x^{3}](https://tex.z-dn.net/?f=V%20%3D%20108%5Ccdot%20x-18%5Ccdot%20x%5E%7B2%7D-24%5Ccdot%20x%5E%7B2%7D%2B4%5Ccdot%20x%5E%7B3%7D)
![V = 108\cdot x -42\cdot x^{2}+4\cdot x^{3}](https://tex.z-dn.net/?f=V%20%3D%20108%5Ccdot%20x%20-42%5Ccdot%20x%5E%7B2%7D%2B4%5Ccdot%20x%5E%7B3%7D)
The polynomial function for the volume of the box (
) in terms of
is
.