To find the volume of the cylinder in terms of [tex]\(b\)[/tex], let's proceed step-by-step using the given relations and the formula for the volume of a right circular cylinder.
1. Given Relations:
- [tex]\( r = 2b \)[/tex]
- [tex]\( h = 5b + 3 \)[/tex]
2. Volume Formula for a Cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
3. Substitute the given values of [tex]\(r\)[/tex] and [tex]\(h\)[/tex] into the volume formula:
[tex]\[
V = \pi (2b)^2 (5b + 3)
\][/tex]
4. Simplify the expression:
[tex]\[
V = \pi (4b^2) (5b + 3)
\][/tex]
5. Distribute [tex]\(4b^2\)[/tex] within the parentheses:
[tex]\[
V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
6. Perform the multiplication:
[tex]\[
V = \pi (20b^3 + 12b^2)
\][/tex]
7. Combine terms:
[tex]\[
V = 20 \pi b^3 + 12 \pi b^2
\][/tex]
Hence, the volume of the cylinder in terms of [tex]\(b\)[/tex] is:
[tex]\[
V = 20 \pi b^3 + 12 \pi b^2
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
\boxed{20 \pi b^3 + 12 \pi b^2}
\][/tex]
