Sure, let's break this down step by step using the formula for the volume of a right circular cylinder, which is given by:
[tex]\[ V = \pi r^2 h \][/tex]
Given:
- [tex]\( r = 2b \)[/tex] (radius in terms of [tex]\( b \)[/tex])
- [tex]\( h = 5b + 3 \)[/tex] (height in terms of [tex]\( b \)[/tex])
Now, let's substitute [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula.
1. Substitute the radius [tex]\( r = 2b \)[/tex] into the formula [tex]\( \pi r^2 h \)[/tex]:
[tex]\[
V = \pi (2b)^2 h
\][/tex]
2. Calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[
(2b)^2 = 4b^2
\][/tex]
So now we have:
[tex]\[
V = \pi (4b^2) h
\][/tex]
3. Substitute the height [tex]\( h = 5b + 3 \)[/tex] into the formula:
[tex]\[
V = \pi (4b^2)(5b + 3)
\][/tex]
4. Distribute [tex]\( 4b^2 \)[/tex] through the term [tex]\( (5b + 3) \)[/tex]:
[tex]\[
V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
5. Perform the multiplications:
[tex]\[
4b^2 \cdot 5b = 20b^3
\][/tex]
[tex]\[
4b^2 \cdot 3 = 12b^2
\][/tex]
6. Add the two products together within the same parentheses:
[tex]\[
V = \pi (20b^3 + 12b^2)
\][/tex]
7. Factor the [tex]\(\pi\)[/tex] back into the equation:
[tex]\[
V = 20\pi b^3 + 12\pi b^2
\][/tex]
So, the volume of the cylinder in terms of [tex]\( b \)[/tex] is [tex]\(\boxed{20 \pi b^3 + 12 \pi b^2}\)[/tex].
That matches with the given choices, so the correct answer from the options provided is:
[tex]\[
\boxed{20 \pi b^3 + 12 \pi b^2}
\][/tex]
