To determine the height of the rectangular prism, we need to recall that the volume of a rectangular prism is calculated by multiplying the base area by the height. This relationship can be written algebraically as:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Given:
- The volume of the rectangular prism is [tex]\( x^3 - 3x^2 + 5x - 3 \)[/tex].
- The area of the base is [tex]\( x^2 - 2 \)[/tex].
We need to find the height, which is given by:
[tex]\[ \text{Height} = \frac{\text{Volume}}{\text{Base Area}} \][/tex]
Substituting the given expressions for Volume and Base Area into the equation, we get:
[tex]\[ \text{Height} = \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
We are given some possible answers in a specific form. To check which one of the given options matches our derived expression, we should manipulate them into a more recognizable form.
Let's simplify each given option to see if it matches our derived expression [tex]\(\frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2}\)[/tex]:
- Option 1: [tex]\(x-3+\frac{7x-9}{x^2-2}\)[/tex]
This option doesn't look like it matches our derived expression, as it has [tex]\(7x - 9\)[/tex] in the numerator of the fraction, which is not correct.
- Option 2: [tex]\(x-3+\frac{7x-9}{x^3-3x^2+5x-3}\)[/tex]
This option also doesn't match, because it has [tex]\(x^3-3x^2+5x-3\)[/tex] in the denominator, which would make it very different from [tex]\(\frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2}\)[/tex].
- Option 3: [tex]\(x-3+\frac{7x+3}{x^2-2}\)[/tex]
This option does not match our derived expression because the numerator of the fraction part [tex]\(7x + 3\)[/tex] doesn’t correspond to our simplified form.
- Option 4: [tex]\(x-3+\frac{7x+3}{x^3-3x^2+5x-3}\)[/tex]
This also doesn’t look correct for similar reasons mentioned above.
None of the given options look exactly like our derived solution. The correct representation of the height of the rectangular prism, given its volume and base area, is:
[tex]\[ \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
Thus, if we simplify any of the given options and compare them with our derived expression, option 3 doesn't precisely match our expression. Given the set of options in the provided question, none of them seem to represent the final height accurately.
The correct height of the rectangular prism that matches our derived expression should be:
[tex]\[ \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]
Considering the options provided do not represent a correct match.
