To find the volume of the cylinder in terms of [tex]\( b \)[/tex], we need to use the given formula for the volume [tex]\( V \)[/tex] of a right circular cylinder, which is:
[tex]\[ V = \pi r^2 h \][/tex]
We are given:
[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/tex]
First, substitute [tex]\( r \)[/tex] into the formula:
[tex]\[ V = \pi (2b)^2 h \][/tex]
Simplify the expression within the parentheses:
[tex]\[ (2b)^2 = 4b^2 \][/tex]
Therefore:
[tex]\[ V = \pi (4b^2) h \][/tex]
Next, substitute [tex]\( h \)[/tex]:
[tex]\[ V = \pi (4b^2) (5b + 3) \][/tex]
Distribute [tex]\( 4b^2 \)[/tex] through [tex]\( (5b + 3) \)[/tex]:
[tex]\[ V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]
Simplifying this expression, we get:
[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]
Among the given choices, the correct one is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
