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Solve the following problem and select your answer from the choices given.

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]

Sagot :

GinnyAnsweridn
To solve the problem, let's start with the given formula for the volume of a right circular cylinder:

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

We are given the following expressions for [tex]\(r\)[/tex] and [tex]\(h\)[/tex]:

[tex]\[ r = 2b \][/tex]
[tex]\[ h = 5b + 3 \][/tex]

We need to substitute these expressions into the volume formula.

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

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

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

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

So, the equation for the volume now looks like this:

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

Now, substitute [tex]\(h = 5b + 3\)[/tex]:

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

Next, distribute [tex]\(4b^2\)[/tex] through the parentheses [tex]\((5b + 3)\)[/tex]:

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

Finally, factor out [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]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]

So, the correct choice from the given options is:

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