- The height \(h(x)\) is found by dividing the volume polynomial \(V(x) = x^4+4x^3+3x^2+8x+4\) by the base area polynomial \(A(x) = x^3+3x^2+8\).
- Using polynomial long division, the division \(V(x) \div A(x)\) yields a quotient of \(x+1\) and a remainder of \(-4\).
- The final expression for the height is the quotient plus the remainder divided by the divisor: \(h(x) = x+1 - \frac{4}{x^3+3x^2+8}\).
- The correct option is: \(\boxed{x+1-\frac{4}{x^3+3x^2+8}}\).
Explanation
- Analysis and Setup
To find the height of the rectangular prism, we use the relationship between volume, base area, and height. Given the volume \(V(x) = x^4+4x^3+3x^2+8x+4\) and the base area \(A(x) = x^3+3x^2+8\), the height \(h(x)\) is defined as the quotient:
\[h(x) = \frac{V(x)}{A(x)} = \frac{x^4+4x^3+3x^2+8x+4}{x^3+3x^2+8}\]
We will perform polynomial long division to simplify this expression.
- Polynomial Long Division
First, we divide the leading term of the numerator (\(x^4\)) by the leading term of the denominator (\(x^3\)):
\[\frac{x^4}{x^3} = x\]
Next, multiply this result by the denominator:
\[x(x^3+3x^2+8) = x^4+3x^3+8x\]
Subtract this from the original numerator:
\[(x^4+4x^3+3x^2+8x+4) - (x^4+3x^3+8x) = x^3+3x^2+4\]
Now, divide the leading term of this new expression (\(x^3\)) by the leading term of the denominator (\(x^3\)):
\[\frac{x^3}{x^3} = 1\]
Multiply this by the denominator:
\[1(x^3+3x^2+8) = x^3+3x^2+8\]
Finally, subtract this from the current remainder:
\[(x^3+3x^2+4) - (x^3+3x^2+8) = -4\]
Since we cannot divide \(-4\) by \(x^3+3x^2+8\) further (as the degree of \(-4\) is less than the degree of the denominator), the remainder is \(-4\).
- Final Conclusion
Combining the quotient and the remainder, we get:
\[h(x) = x + 1 + \frac{-4}{x^3+3x^2+8}\]
This can be rewritten as:
\[h(x) = x + 1 - \frac{4}{x^3+3x^2+8}\]
Comparing this result with the provided options, it matches the third option. Therefore, the height of the prism is \(\boxed{x+1-\frac{4}{x^3+3x^2+8}}\).
Examples
In engineering and architecture, understanding the dimensions of structures is vital. Suppose you are designing a storage box and you know the total volume and the base dimensions required to fit your items. By using polynomial division, you can calculate the necessary height to ensure that the box perfectly holds the intended volume, allowing for precise material planning and efficient design. This mathematical approach helps in optimizing storage capacity and material usage in real-world manufacturing and construction tasks.