Discover how IDNLearn.com can help you find the answers you need quickly and easily. Ask any question and receive accurate, in-depth responses from our dedicated team of experts.
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]
[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]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.