To find the volume of a right circular cylinder given [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex], we start with the formula for the volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
First, substitute [tex]\( r = 2b \)[/tex] into the formula:
[tex]\[ V = \pi (2b)^2 h \][/tex]
Simplify [tex]\( (2b)^2 \)[/tex]:
[tex]\[ (2b)^2 = 4b^2 \][/tex]
Thus, the volume formula now becomes:
[tex]\[ V = \pi (4b^2) h \][/tex]
Next, substitute [tex]\( h = 5b + 3 \)[/tex] into the formula:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]
Now, distribute [tex]\( 4b^2 \)[/tex] in the expression:
[tex]\[ V = \pi (4b^2) (5b) + \pi (4b^2) (3) \][/tex]
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/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]
Among the given choices, the correct answer is:
[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]
