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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 :

GinnyAnswer
Sure, let's walk through the problem step-by-step to find the volume [tex]\( V \)[/tex] of the cylinder in terms of [tex]\( b \)[/tex].

We know:
- The formula for the volume of a right circular cylinder is [tex]\( V = \pi r^2 h \)[/tex].
- The radius [tex]\( r \)[/tex] is given as [tex]\( r = 2b \)[/tex].
- The height [tex]\( h \)[/tex] is given as [tex]\( h = 5b + 3 \)[/tex].

To find the volume of the cylinder in terms of [tex]\( b \)[/tex]:

1. Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:

[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]

2. Simplify the expression inside the parentheses:

[tex]\[ (2b)^2 = 4b^2 \][/tex]

So the volume formula becomes:

[tex]\[ V = \pi (4b^2)(5b + 3) \][/tex]

3. Distribute [tex]\( 4b^2 \)[/tex] across the terms inside the parentheses:

[tex]\[ V = \pi (4b^2 \cdot 5b + 4b^2 \cdot 3) \][/tex]

[tex]\[ V = \pi (20b^3 + 12b^2) \][/tex]

4. Factor out the common terms:

[tex]\[ V = 4\pi b^2 (5b + 3) \][/tex]

Given this simplified expression, we have:

- [tex]\( V = 4\pi b^2 (5b + 3) \)[/tex]

The correct answer in terms of [tex]\( b \)[/tex] is:

[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]

So, the corresponding option is

[tex]\[ \boxed{20\pi b^3 + 12\pi b^2} \][/tex]

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