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][tex]$20 \pi^2 b^3+12 \pi^2 b^2$[/tex][/tex]
D. [tex]$50 \pi b^3+20 \pi b^2+90 \pi b$[/tex]



Answer :

To find the volume of a right circular cylinder given [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex], we start with the formula for the volume of a cylinder:

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

First, substitute [tex]\( r = 2b \)[/tex] into the formula:

[tex]\[ V = \pi (2b)^2 h \][/tex]

Simplify [tex]\( (2b)^2 \)[/tex]:

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

Thus, the volume formula now becomes:

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

Next, substitute [tex]\( h = 5b + 3 \)[/tex] into the formula:

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

Now, distribute [tex]\( 4b^2 \)[/tex] in the expression:

[tex]\[ V = \pi (4b^2) (5b) + \pi (4b^2) (3) \][/tex]
[tex]\[ V = 20\pi b^3 + 12\pi b^2 \][/tex]

Therefore, the volume of the cylinder in terms of [tex]\( b \)[/tex] is:

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

Among the given choices, the correct answer is:

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