To find the volume of a right circular cylinder in terms of the variable [tex]\( b \)[/tex], we start with the formula for the volume of a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
Given that [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex], we substitute these expressions into the formula.
First, we substitute [tex]\( r = 2b \)[/tex] into the equation:
[tex]\[
r^2 = (2b)^2 = 4b^2
\][/tex]
Next, substitute both [tex]\( r^2 \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[
V = \pi (r^2) h = \pi (4b^2)(5b + 3)
\][/tex]
Now, distribute [tex]\( 4b^2 \)[/tex] across [tex]\( 5b + 3 \)[/tex]:
[tex]\[
V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
[tex]\[
V = \pi (20b^3 + 12b^2)
\][/tex]
Therefore, the volume of the cylinder in terms of [tex]\( b \)[/tex] can be written as:
[tex]\[
V = 20\pi b^3 + 12\pi b^2
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{20 \pi b^3 + 12 \pi b^2}
\][/tex]
