The formula for the volume of a right circular cylinder is [tex] V = \pi r^2 h [/tex]. If [tex] r = 2b [/tex] and [tex] h = 5b + 3 [/tex], what is the volume of the cylinder in terms of [tex] b [/tex]?

A. [tex] 10\pi b^2 + 6\pi b [/tex]
B. [tex] 20\pi b^3 + 12\pi b^2 [/tex]
C. [tex] 20\pi^2 b^3 + 12\pi^2 b^2 [/tex]
D. [tex] 50\pi b^3 + 20\pi b^2 + 90\pi b [/tex]



Answer :

To find the volume of the cylinder in terms of [tex]\( b \)[/tex], let's follow a step-by-step procedure using the given formulas for [tex]\( r \)[/tex] and [tex]\( h \)[/tex].

1. Identify the volume formula:
The volume [tex]\( V \)[/tex] of a right circular cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]

2. Substitute the expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
Given:
[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/tex]

3. Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]

4. Simplify the expression:
- First, calculate [tex]\( (2b)^2 \)[/tex]:
[tex]\[ (2b)^2 = (2b) \cdot (2b) = 4b^2 \][/tex]
- So, the volume formula now becomes:
[tex]\[ V = \pi \cdot 4b^2 \cdot (5b + 3) \][/tex]

5. Expand the expression inside the parentheses:
- Distribute [tex]\( 4b^2 \)[/tex] into [tex]\( (5b + 3) \)[/tex]:
[tex]\[ V = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]
- Calculate each term separately:
[tex]\[ 4b^2 \cdot 5b = 20 b^3 \][/tex]
[tex]\[ 4b^2 \cdot 3 = 12 b^2 \][/tex]

6. Combine the terms:
[tex]\[ V = \pi (20 b^3 + 12 b^2) \][/tex]
- Distribute [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]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{20 \pi b^3 + 12 \pi b^2} \][/tex]