The formula for the volume of a right circular cylinder is [tex]\( V = \pi r^2 h \)[/tex].

If [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex], what is the volume of the cylinder in terms of [tex]\( b \)[/tex]?

A. [tex]\( 10 \pi b^2 + 6 \pi b \)[/tex]
B. [tex]\( 20 \pi b^3 + 12 \pi b^2 \)[/tex]
C. [tex]\( 20 \pi^2 b^3 + 12 \pi^2 b^2 \)[/tex]
D. [tex]\( 50 \pi b^3 + 20 \pi b^2 + 90 \pi b \)[/tex]



Answer :

The problem requires finding the volume of a right circular cylinder expressed in terms of [tex]\( b \)[/tex]. We will use the formula for the volume of a cylinder, which is:
[tex]\[ V = \pi r^2 h \][/tex]

Given:
- [tex]\( r = 2b \)[/tex]
- [tex]\( h = 5b + 3 \)[/tex]

We will substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula.

1. Substitute [tex]\( r = 2b \)[/tex] into the formula [tex]\( V = \pi r^2 h \)[/tex]:
[tex]\[ r^2 = (2b)^2 = 4b^2 \][/tex]

2. Next, substitute [tex]\( r^2 \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]

3. Distribute [tex]\( \pi 4b^2 \)[/tex] across the terms inside the parentheses:
[tex]\[ V = \pi \cdot 4b^2 \cdot (5b + 3) \][/tex]
[tex]\[ V = \pi \cdot 4b^2 \cdot 5b + \pi \cdot 4b^2 \cdot 3 \][/tex]
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]

The volume of the cylinder in terms of [tex]\( b \)[/tex] is:
[tex]\[ V = 20 \pi b^3 + 12 \pi b^2 \][/tex]

Thus, the correct answer is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]