To determine the volume of the cylinder given the relationships involving [tex]\( b \)[/tex], let's proceed step-by-step.
1. Recall the formula for the volume of a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( V \)[/tex] is the volume, [tex]\( r \)[/tex] is the radius, and [tex]\( h \)[/tex] is the height.
2. Substitute the expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
The radius [tex]\( r \)[/tex] is given by:
[tex]\[
r = 2b
\][/tex]
The height [tex]\( h \)[/tex] is given by:
[tex]\[
h = 5b + 3
\][/tex]
3. Plug these values into the formula:
Substitute [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula:
[tex]\[
V = \pi (2b)^2 (5b + 3)
\][/tex]
4. Simplify the expression:
First, calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[
(2b)^2 = 4b^2
\][/tex]
Now substitute [tex]\( 4b^2 \)[/tex] and [tex]\( 5b + 3 \)[/tex] back into the volume formula:
[tex]\[
V = \pi (4b^2) (5b + 3)
\][/tex]
5. Distribute [tex]\( 4b^2 \)[/tex] inside the parentheses:
Apply the distributive property:
[tex]\[
V = \pi \cdot 4b^2 \cdot (5b + 3) = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3)
\][/tex]
6. Simplify further:
Calculate each term inside the parentheses:
[tex]\[
4b^2 \cdot 5b = 20b^3
\][/tex]
[tex]\[
4b^2 \cdot 3 = 12b^2
\][/tex]
Therefore:
[tex]\[
V = \pi (20b^3 + 12b^2)
\][/tex]
7. Combine the terms to get the final expression:
[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]\[
V = 20\pi b^3 + 12\pi b^2
\][/tex]
