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Sagot :
To solve the problem of finding the height of the rectangular prism, we start by applying the formula for the volume of a rectangular prism, which is given by the product of its base area and height:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Given:
- The volume of the rectangular prism is [tex]\( V = x^4 + 4x^3 + 3x^2 + 8x + 4 \)[/tex]
- The area of the base of the rectangular prism is [tex]\( A = x^3 + 3x^2 + 8 \)[/tex]
We need to determine the height [tex]\( h \)[/tex] of the prism. We can derive the height by rearranging the volume formula to solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{V}{A} \][/tex]
Substitute the expressions for the volume and base area:
[tex]\[ h = \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \][/tex]
Next, we simplify the expression [tex]\( \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \)[/tex]:
To simplify this rational function, consider polynomial long division:
1. The leading term of the numerator [tex]\( x^4 \)[/tex] divided by the leading term of the denominator [tex]\( x^3 \)[/tex]:
[tex]\[ x \][/tex]
2. Multiply the entire divisor [tex]\( x^3 + 3x^2 + 8 \)[/tex] by this quotient:
[tex]\[ x(x^3 + 3x^2 + 8) = x^4 + 3x^3 + 8x \][/tex]
3. Subtract this product from the numerator:
[tex]\[ (x^4 + 4x^3 + 3x^2 + 8x + 4) - (x^4 + 3x^3 + 8x) = x^4 + 4x^3 + 3x^2 + 8x + 4 - x^4 - 3x^3 - 8x = x^3 + 3x^2 + 4 \][/tex]
4. The new expression becomes:
[tex]\[ \frac{x^3 + 3x^2 + 4}{x^3 + 3x^2 + 8} \][/tex]
We notice that by performing polynomial division, we have:
[tex]\[ x^3 + 3x^2 + 4 \][/tex]
This shows a quotient simplification leading to the final result, combined with the original leading term quotient [tex]\( x \)[/tex]:
Now we test against the options to match it properly.
Upon simplifying and matching with the given options:
- [tex]\( x+1-\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1+\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1-\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]
- [tex]\( x+1+\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]
We find that:
[tex]\[ \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} = x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]
Thus, the height of the prism is:
[tex]\[ \boxed{3} \][/tex]
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Given:
- The volume of the rectangular prism is [tex]\( V = x^4 + 4x^3 + 3x^2 + 8x + 4 \)[/tex]
- The area of the base of the rectangular prism is [tex]\( A = x^3 + 3x^2 + 8 \)[/tex]
We need to determine the height [tex]\( h \)[/tex] of the prism. We can derive the height by rearranging the volume formula to solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{V}{A} \][/tex]
Substitute the expressions for the volume and base area:
[tex]\[ h = \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \][/tex]
Next, we simplify the expression [tex]\( \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} \)[/tex]:
To simplify this rational function, consider polynomial long division:
1. The leading term of the numerator [tex]\( x^4 \)[/tex] divided by the leading term of the denominator [tex]\( x^3 \)[/tex]:
[tex]\[ x \][/tex]
2. Multiply the entire divisor [tex]\( x^3 + 3x^2 + 8 \)[/tex] by this quotient:
[tex]\[ x(x^3 + 3x^2 + 8) = x^4 + 3x^3 + 8x \][/tex]
3. Subtract this product from the numerator:
[tex]\[ (x^4 + 4x^3 + 3x^2 + 8x + 4) - (x^4 + 3x^3 + 8x) = x^4 + 4x^3 + 3x^2 + 8x + 4 - x^4 - 3x^3 - 8x = x^3 + 3x^2 + 4 \][/tex]
4. The new expression becomes:
[tex]\[ \frac{x^3 + 3x^2 + 4}{x^3 + 3x^2 + 8} \][/tex]
We notice that by performing polynomial division, we have:
[tex]\[ x^3 + 3x^2 + 4 \][/tex]
This shows a quotient simplification leading to the final result, combined with the original leading term quotient [tex]\( x \)[/tex]:
Now we test against the options to match it properly.
Upon simplifying and matching with the given options:
- [tex]\( x+1-\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1+\frac{4}{x^3+3 x^2+8} \)[/tex]
- [tex]\( x+1-\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]
- [tex]\( x+1+\frac{4}{x^4+4 x^3+3 x^2+8 x+4} \)[/tex]
We find that:
[tex]\[ \frac{x^4 + 4x^3 + 3x^2 + 8x + 4}{x^3 + 3x^2 + 8} = x + 1 - \frac{4}{x^3 + 3x^2 + 8} \][/tex]
Thus, the height of the prism is:
[tex]\[ \boxed{3} \][/tex]
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