Answered

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Question:

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

Sagot :

GinnyAnswer
Sure! Let's solve this step-by-step.

We start with the formula for the volume of a right circular cylinder:

[tex]\[ V = \pi r^2 h \][/tex]

Given values are:
- [tex]\( r = 2b \)[/tex]
- [tex]\( h = 5b + 3 \)[/tex]

First, substitute [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula:

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

Next, calculate [tex]\( (2b)^2 \)[/tex]:

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

Now, substitute this back into the volume formula:

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

Distribute [tex]\( 4b^2 \)[/tex] through the terms inside the parenthesis:

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

Perform the multiplication:

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

Therefore, the correct answer is:

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