To determine which of the two given expressions correctly represents the volume of a cylinder, we need to compare them with the standard volume formula for a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Given the expressions:
1. [tex]\( 2 \pi x^3 - 12 \pi x^2 - 24 \pi x + 63 \pi \)[/tex]
2. [tex]\( 2 \pi x^3 - 5 \pi x^2 - 24 \pi x + 63 \pi \)[/tex]
Let's break down the expressions term by term and see which aligns more logically with the formula [tex]\( V = \pi r^2 h \)[/tex].
### Key Observations:
1. The standard volume formula involves a product of [tex]\(\pi\)[/tex], [tex]\(r^2\)[/tex] (square of the radius), and [tex]\(h\)[/tex] (height). Thus, we need polynomial terms and coefficients that can reasonably relate to this form.
2. Both given expressions are polynomial in terms of [tex]\(x\)[/tex], including the factor [tex]\(\pi\)[/tex], which is consistent with the use of [tex]\(\pi\)[/tex] in the formula.
### Expression Comparison:
First Expression: [tex]\(2 \pi x^3 - 12 \pi x^2 - 24 \pi x + 63 \pi\)[/tex]
- Here, the leading term is [tex]\(2 \pi x^3\)[/tex], which suggests a dimension of [tex]\(x^3\)[/tex].
- Other terms include [tex]\( - 12 \pi x^2\)[/tex], [tex]\( - 24 \pi x\)[/tex], and a constant term [tex]\(63 \pi\)[/tex].
Second Expression: [tex]\(2 \pi x^3 - 5 \pi x^2 - 24 \pi x + 63 \pi\)[/tex]
- Similarly, the leading term is [tex]\(2 \pi x^3\)[/tex].
- Other terms include [tex]\( - 5 \pi x^2\)[/tex], [tex]\( - 24 \pi x\)[/tex], and again a constant term [tex]\(63 \pi\)[/tex].
### Conclusion:
The correct expression for the volume of the cylinder using the formula [tex]\( V = \pi r^2 h \)[/tex] should factor correctly and consistently into this mathematical configuration.
After evaluating the two expressions given, we determine that the expression that aligns more closely with the characteristics of a well-formed polynomial representing the volume of an actual cylinder from the given options:
1. [tex]\(2 \pi x^3 - 12 \pi x^2 - 24 \pi x + 63 \pi\)[/tex]
Thus, the first expression:
[tex]\[ 2 \pi x^3 - 12 \pi x^2 - 24 \pi x + 63 \pi \][/tex]
represents the volume of the cylinder.
