To find the volume of the cylinder in terms of [tex]\( b \)[/tex], let's follow a step-by-step procedure using the given formulas for [tex]\( r \)[/tex] and [tex]\( h \)[/tex].
1. Identify the volume formula:
The volume [tex]\( V \)[/tex] of a right circular cylinder is given by:
[tex]\[
V = \pi r^2 h
\][/tex]
2. Substitute the expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
Given:
[tex]\[
r = 2b
\][/tex]
[tex]\[
h = 5b + 3
\][/tex]
3. Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[
V = \pi (2b)^2 (5b + 3)
\][/tex]
4. Simplify the expression:
- First, calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[
(2b)^2 = (2b) \cdot (2b) = 4b^2
\][/tex]
- So, the volume formula now becomes:
[tex]\[
V = \pi \cdot 4b^2 \cdot (5b + 3)
\][/tex]
5. Expand the expression inside the parentheses:
- Distribute [tex]\( 4b^2 \)[/tex] into [tex]\( (5b + 3) \)[/tex]:
[tex]\[
V = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
- Calculate each term separately:
[tex]\[
4b^2 \cdot 5b = 20 b^3
\][/tex]
[tex]\[
4b^2 \cdot 3 = 12 b^2
\][/tex]
6. Combine the terms:
[tex]\[
V = \pi (20 b^3 + 12 b^2)
\][/tex]
- Distribute [tex]\(\pi\)[/tex]:
[tex]\[
V = 20 \pi b^3 + 12 \pi b^2
\][/tex]
Thus, the volume of the cylinder in terms of [tex]\( b \)[/tex] is:
[tex]\[
\boxed{20 \pi b^3 + 12 \pi b^2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{20 \pi b^3 + 12 \pi b^2}
\][/tex]
