Sure, let's walk through the problem step-by-step to find the volume [tex]\( V \)[/tex] of the cylinder in terms of [tex]\( b \)[/tex].
We know:
- The formula for the volume of a right circular cylinder is [tex]\( V = \pi r^2 h \)[/tex].
- The radius [tex]\( r \)[/tex] is given as [tex]\( r = 2b \)[/tex].
- The height [tex]\( h \)[/tex] is given as [tex]\( h = 5b + 3 \)[/tex].
To find the volume of the cylinder in terms of [tex]\( b \)[/tex]:
1. Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[
V = \pi (2b)^2 (5b + 3)
\][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[
(2b)^2 = 4b^2
\][/tex]
So the volume formula becomes:
[tex]\[
V = \pi (4b^2)(5b + 3)
\][/tex]
3. Distribute [tex]\( 4b^2 \)[/tex] across the terms inside the parentheses:
[tex]\[
V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
[tex]\[
V = \pi (20b^3 + 12b^2)
\][/tex]
4. Factor out the common terms:
[tex]\[
V = 4\pi b^2 (5b + 3)
\][/tex]
Given this simplified expression, we have:
- [tex]\( V = 4\pi b^2 (5b + 3) \)[/tex]
The correct answer in terms of [tex]\( b \)[/tex] is:
[tex]\[
V = 20\pi b^3 + 12\pi b^2
\][/tex]
So, the corresponding option is
[tex]\[ \boxed{20\pi b^3 + 12\pi b^2} \][/tex]
