To solve this problem, we need to use our understanding of the geometric properties of a cylinder, as well as how those properties are represented in the given formula for volume.
First, let's recall the volume formula for a cylinder, which is:
[tex]\[
V = \pi r^2 h
\][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height,
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159).
In the given formula for the volume of the cylindrical soap dispenser:
[tex]\[
\pi(x+1)^2(4x+1)
\][/tex]
we need to identify which parts of the expression correspond to the radius and the height of the cylinder.
The expression [tex]\(\pi(x+1)^2(4x+1)\)[/tex] is structured to follow the pattern of [tex]\(\pi r^2 h\)[/tex]:
1. The [tex]\(\pi(x+1)^2\)[/tex] part appears to represent [tex]\(\pi r^2\)[/tex]. Here, [tex]\((x+1)\)[/tex] is inside the square, indicating that [tex]\((x+1)\)[/tex] represents the radius [tex]\( r \)[/tex] of the cylinder.
- So, [tex]\((x+1)^2\)[/tex] is [tex]\( r^2 \)[/tex] and multiplying by [tex]\(\pi\)[/tex] we get [tex]\(\pi r^2\)[/tex], which is the base area of the cylinder.
2. That leaves the [tex]\((4x+1)\)[/tex] part of the expression. Since [tex]\( \pi (x+1)^2 \)[/tex] represents the base area [tex]\( (\pi r^2) \)[/tex], the remaining factor, [tex]\( (4x+1) \)[/tex], must represent the height [tex]\( h \)[/tex] of the cylinder.
Therefore, analyzing the components of the given expression:
- The expression [tex]\(\pi(x+1)^2\)[/tex] represents the area of the base.
- The factor [tex]\(4x+1\)[/tex] represents the height.
Given these observations, the correct statement about the given expression is:
B. The factor [tex]\(4 x+1\)[/tex] represents the height of the soap dispenser.
