To determine the height of the rectangular prism given the volume [tex]\(V\)[/tex] and the base area [tex]\(A\)[/tex], we need to express [tex]\(V\)[/tex] in terms of [tex]\(A\)[/tex] and solve for the height [tex]\(h\)[/tex].
Given:
- Volume, [tex]\(V = x^4 + 4x^3 + 3x^2 + 8x + 4\)[/tex]
- Base area, [tex]\(A = x^3 + 3x^2 + 8\)[/tex]
We know:
[tex]\[ V = A \cdot h \][/tex]
Therefore:
[tex]\[ h = \frac{V}{A} \][/tex]
First, let's perform the polynomial division of [tex]\(V\)[/tex] by [tex]\(A\)[/tex].
Step-by-step polynomial division:
1. Divide the leading term of [tex]\(V\)[/tex] by the leading term of [tex]\(A\)[/tex]:
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
So, the first term of the quotient (height) is [tex]\(x\)[/tex].
2. Multiply [tex]\(A\)[/tex] by [tex]\(x\)[/tex] and subtract from [tex]\(V\)[/tex]:
[tex]\[ x \cdot (x^3 + 3x^2 + 8) = x^4 + 3x^3 + 8x \][/tex]
[tex]\[ (x^4 + 4x^3 + 3x^2 + 8x + 4) - (x^4 + 3x^3 + 8x) = x^3 + 3x^2 + 4 \][/tex]
3. Divide the new leading term [tex]\(x^3\)[/tex] by the leading term of [tex]\(A\)[/tex]:
[tex]\[ \frac{x^3}{x^3} = 1 \][/tex]
So, the next term in the quotient is [tex]\(1\)[/tex].
4. Multiply [tex]\(A\)[/tex] by [tex]\(1\)[/tex] and subtract from the new polynomial:
[tex]\[ 1 \cdot (x^3 + 3x^2 + 8) = x^3 + 3x^2 + 8 \][/tex]
[tex]\[ (x^3 + 3x^2 + 4) - (x^3 + 3x^2 + 8) = 4 - 8 = -4 \][/tex]
Thus, after the polynomial division we get:
[tex]\[ h = x + 1 + \frac{-4}{x^3 + 3x^2 + 8} \][/tex]
or equivalently,
[tex]\[ h = x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]
Therefore, the height of the prism is:
[tex]\[ h = x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]
The correct answer is:
[tex]\[ x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]
