Sure! Let's solve this step-by-step.
We start with the formula for the volume of a right circular cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Given values are:
- [tex]\( r = 2b \)[/tex]
- [tex]\( h = 5b + 3 \)[/tex]
First, substitute [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula:
[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]
Next, calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[ (2b)^2 = (2b)(2b) = 4b^2 \][/tex]
Now, substitute this back into the volume formula:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]
Distribute [tex]\( 4b^2 \)[/tex] through the terms inside the parenthesis:
[tex]\[ V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
Perform the multiplication:
[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]
Thus, 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 is:
[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]
