Let's start by substituting the given expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the formula for the volume of a right circular cylinder.
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[ V = \pi r^2 h \][/tex]
We have:
[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/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 becomes:
[tex]\[ V = \pi \cdot 4b^2 \cdot h \][/tex]
Next, substitute [tex]\( h = 5b + 3 \)[/tex] into the volume formula:
[tex]\[ V = \pi \cdot 4b^2 \cdot (5b + 3) \][/tex]
Distribute [tex]\( 4b^2 \)[/tex] within the parentheses:
[tex]\[ V = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
[tex]\[ V = \pi \cdot (20b^3 + 12b^2) \][/tex]
Distribute [tex]\(\pi\)[/tex] throughout the terms:
[tex]\[ V = 20 \pi b^3 + 12 \pi b^2 \][/tex]
Therefore, the volume [tex]\( V \)[/tex] of the cylinder in terms of [tex]\( b \)[/tex] is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
Hence, the correct answer is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
