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The volume of a cylinder is given by the formula [tex]$V = \pi r^2 h$[/tex], where [tex]$r$[/tex] is the radius of the cylinder and [tex][tex]$h$[/tex][/tex] is the height.

Which expression represents the volume of this cylinder?

A. [tex]$2 \pi x^3 - 12 \pi x^2 - 24 \pi x + 63 \pi$[/tex]

B. [tex]$2 \pi x^3 - 5 \pi x^2 - 24 \pi x + 63 \pi$[/tex]

Sagot :

GinnyAnsweridn
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.
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