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]
