To find the volume of a cylinder given the radius [tex]\( r = 2b \)[/tex] and height [tex]\( h = 5b + 3 \)[/tex], we'll use the formula for the volume of a right circular cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Step-by-step, we can substitute the given expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the formula:
1. Substitute [tex]\( r = 2b \)[/tex]:
[tex]\[ r^2 = (2b)^2 = 4b^2 \][/tex]
2. Substitute [tex]\( r^2 = 4b^2 \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]
3. Distribute [tex]\( 4b^2 \)[/tex] over [tex]\( 5b + 3 \)[/tex]:
[tex]\[ 4b^2 \cdot 5b + 4b^2 \cdot 3 = 20b^3 + 12b^2 \][/tex]
4. Multiply by [tex]\(\pi\)[/tex]:
[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]
So, the volume of the cylinder in terms of [tex]\( b \)[/tex] is:
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]
The correct choice is:
[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]
