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The volume of a cylinder is given by [tex][tex]$v=\pi r^2 h$[/tex][/tex], where [tex][tex]$r$[/tex][/tex] is the radius of the cylinder and [tex][tex]$h$[/tex][/tex] is the height. Which expression represents the volume of this can?

A. [tex][tex]$3 \pi x^2 + 4 \pi x + 16 \pi$[/tex][/tex]
B. [tex][tex]$3 \pi x^2 + 16 \pi$[/tex][/tex]
C. [tex][tex]$3 \pi x^3 + 32 \pi$[/tex][/tex]
D. [tex][tex]$3 \pi x^3 + 20 \pi x^2 + 44 \pi x + 32 \pi$[/tex][/tex]

Sagot :

To determine which of the given expressions represents the volume of a cylinder, we must recall the formula for the volume of a cylinder. The volume [tex]\( V \)[/tex] of a cylinder with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by:

[tex]\[ V = \pi r^2 h \][/tex]

In this context, suppose the radius [tex]\( r \)[/tex] is denoted by [tex]\( x \)[/tex] and the height [tex]\( h \)[/tex] is also denoted by [tex]\( x \)[/tex]. Then, the volume formula becomes:

[tex]\[ V = \pi x^2 \cdot x \][/tex]
[tex]\[ V = \pi x^3 \][/tex]

We need to identify which of the given expressions matches the form [tex]\( \pi x^3 \)[/tex].

1. [tex]\( 3 \pi x^2 + 4 \pi x + 16 \pi \)[/tex]
- This expression does not match [tex]\( \pi x^3 \)[/tex]; it contains terms with [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and a constant term.

2. [tex]\( 3 \pi x^2 + 16 \pi \)[/tex]
- This expression also does not match [tex]\( \pi x^3 \)[/tex]; it consists of terms with [tex]\( x^2 \)[/tex] and a constant.

3. [tex]\( 3 \pi x^3 + 32 \pi \)[/tex]
- This expression includes a term [tex]\( 3 \pi x^3 \)[/tex] plus a constant term. Notice that the coefficient of [tex]\( x^3 \)[/tex] is 3, not 1. While it contains [tex]\( \pi x^3 \)[/tex], it is actually multiplied by 3.

4. [tex]\( 3 \pi x^3 + 20 \pi x^2 + 44 \pi x + 32 \pi \)[/tex]
- This expression contains multiple terms with different powers of [tex]\( x \)[/tex] and does not match the simple form [tex]\( \pi x^3 \)[/tex].

Among the given expressions, option 3, [tex]\( 3 \pi x^3 + 32 \pi \)[/tex], includes the term [tex]\( 3 \pi x^3 \)[/tex]. It suggests a connection to the volume formula [tex]\( \pi x^3 \)[/tex], albeit scaled by a constant.

Thus, the expression that includes the term matching the volume formula under given assumptions is:

[tex]\[ 3 \pi x^3 + 32 \pi \][/tex]

Therefore, the correct answer is:

[tex]\[ 3 \pi x^3 + 32 \pi \][/tex]

Which corresponds to the third option.