To find the volume of a right circular cylinder in terms of [tex]\( b \)[/tex], we start with the formula for the volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Given:
[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/tex]
First, substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]
Calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[ (2b)^2 = 4b^2 \][/tex]
Now, substitute [tex]\( 4b^2 \)[/tex] back into the volume formula:
[tex]\[ V = \pi (4b^2)(5b + 3) \][/tex]
Distribute [tex]\( 4b^2 \)[/tex] in the expression:
[tex]\[ V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
Calculate the terms:
[tex]\[ 4b^2 \cdot 5b = 20b^3 \][/tex]
[tex]\[ 4b^2 \cdot 3 = 12b^2 \][/tex]
Combine the terms:
[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]
Therefore, the volume of the cylinder in terms of [tex]\( b \)[/tex] is:
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]
So, the correct answer is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
This corresponds to the choice:
[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]
