Final answer:
To find the expressions for the height and length of a rectangular box given the volume function and width, perform polynomial division by dividing the volume function by the width. Factor the result to obtain the product of length and height, from which individual expressions can be identified based on standard geometric dimensions.
Step-by-step explanation:
The student is asking to find the expressions for the height and length of a rectangular box with a given volume function V = 2x^3 - 11x^2 + 10x + 8 and a width represented by x - 4. To find the other dimensions, we need to factor the volume function and separate it into the product of the width, length, and height.
Firstly, we assume that the volume V of the box can be represented as V = length × width × height. Given that the width is x - 4, we need to divide the volume function by this expression to find the product of length and height:
V / width = (2x^3 - 11x^2 + 10x + 8) / (x - 4)
Upon performing this division (using long division or polynomial division), we would obtain a quadratic expression, which represents the product of the length and height of the box. By matching it with standard geometric dimensions, we can then find expressions for the individual dimensions of length and height.
For instance, if the obtained expression was A(x) and it can be factorized into (x - a)(x - b), we could then say the length is (x - a) and the height is (x - b), or vice versa, since the width, length, and height are interchangeable in a product representing a volume.