To solve the problem, let's start with the given formula for the volume of a right circular cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
We are given the following expressions for [tex]\(r\)[/tex] and [tex]\(h\)[/tex]:
[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/tex]
We need to substitute these expressions into the volume formula.
First, substitute [tex]\(r = 2b\)[/tex] into the equation:
[tex]\[ V = \pi (2b)^2 h \][/tex]
Next, simplify [tex]\((2b)^2\)[/tex]:
[tex]\[ (2b)^2 = 4b^2 \][/tex]
So, the equation for the volume now looks like this:
[tex]\[ V = \pi \cdot 4b^2 \cdot h \][/tex]
Now, substitute [tex]\(h = 5b + 3\)[/tex]:
[tex]\[ V = \pi \cdot 4b^2 \cdot (5b + 3) \][/tex]
Next, distribute [tex]\(4b^2\)[/tex] through the parentheses [tex]\((5b + 3)\)[/tex]:
[tex]\[ V = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
[tex]\[ V = \pi \cdot (20b^3 + 12b^2) \][/tex]
Finally, factor out [tex]\(\pi\)[/tex]:
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]
Thus, the volume of the cylinder in terms of [tex]\(b\)[/tex] is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
So, the correct choice from the given options is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
