Connect with a global community of knowledgeable individuals on IDNLearn.com. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.

The volume of a rectangular prism is [tex]\(x^3-3x^2+5x-3\)[/tex], and the area of its base is [tex]\(x^2-2\)[/tex]. If the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?

A. [tex]\(x-3+\frac{7x-9}{x^2-2}\)[/tex]

B. [tex]\(x-3+\frac{7x-9}{x^3-3x^2+5x-3}\)[/tex]

C. [tex]\(x-3+\frac{7x+3}{x^2-2}\)[/tex]

D. [tex]\(x-3+\frac{7x+3}{x^3-3x^2+5x-3}\)[/tex]


Sagot :

To determine the height [tex]\( h \)[/tex] of the rectangular prism given its volume and the area of its base, we use the formula for the volume of a prism, which is given by:

[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]

Given:
- The volume [tex]\( V \)[/tex] of the rectangular prism: [tex]\( V = x^3 - 3x^2 + 5x - 3 \)[/tex]
- The area of the base [tex]\( A \)[/tex]: [tex]\( A = x^2 - 2 \)[/tex]

To find the height [tex]\( h \)[/tex], we solve for [tex]\( h \)[/tex] in the volume formula:

[tex]\[ h = \frac{V}{A} = \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} \][/tex]

We can perform polynomial long division for the given polynomial expression:

[tex]\[ x^3 - 3x^2 + 5x - 3 \div x^2 - 2 \][/tex]

1. Divide the leading term of the numerator [tex]\( x^3 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex] to get [tex]\( x \)[/tex].

2. Multiply [tex]\( x \)[/tex] by the denominator [tex]\( x^2 - 2 \)[/tex] to get [tex]\( x^3 - 2x \)[/tex].

3. Subtract [tex]\( x^3 - 2x \)[/tex] from [tex]\( x^3 - 3x^2 + 5x - 3 \)[/tex]:
[tex]\[ (x^3 - 3x^2 + 5x - 3) - (x^3 - 2x) = -3x^2 + 7x - 3 \][/tex]

4. Divide the leading term of the new numerator [tex]\( -3x^2 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex] to get [tex]\( -3 \)[/tex].

5. Multiply [tex]\( -3 \)[/tex] by the denominator [tex]\( x^2 - 2 \)[/tex] to get [tex]\( -3x^2 + 6 \)[/tex].

6. Subtract [tex]\( -3x^2 + 6 \)[/tex] from [tex]\( -3x^2 + 7x - 3 \)[/tex]:
[tex]\[ (-3x^2 + 7x - 3) - (-3x^2 + 6) = 7x - 9 \][/tex]

Thus, the polynomial long division gives us:
[tex]\[ \frac{x^3 - 3x^2 + 5x - 3}{x^2 - 2} = x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]

Therefore, the height of the prism is:
[tex]\[ x - 3 + \frac{7x - 9}{x^2 - 2} \][/tex]

The correct answer is:

[tex]\[ \boxed{x-3+\frac{7 x-9}{x^2-2}} \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.