To solve the problem of finding the volume of a right circular cylinder in terms of [tex]\( b \)[/tex], given [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex], we can use the volume formula of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Step-by-step, we will substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into this formula.
1. Substitute [tex]\( r = 2b \)[/tex] and [tex]\( h = 5b + 3 \)[/tex] into the volume formula:
[tex]\[ V = \pi (2b)^2 (5b + 3) \][/tex]
2. First, calculate the value of [tex]\( r^2 \)[/tex]:
[tex]\[ (2b)^2 = 4b^2 \][/tex]
Now substitute this back into the volume formula:
[tex]\[ V = \pi \cdot 4b^2 \cdot (5b + 3) \][/tex]
3. Distribute [tex]\( 4b^2 \)[/tex] into the expression [tex]\( 5b + 3 \)[/tex]:
[tex]\[ V = \pi \cdot 4b^2 \cdot 5b + \pi \cdot 4b^2 \cdot 3 \][/tex]
[tex]\[ V = \pi (4b^2 \cdot 5b) + \pi (4b^2 \cdot 3) \][/tex]
[tex]\[ V = \pi (20b^3) + \pi (12b^2) \][/tex]
4. Combine the terms:
[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]\[ V = 20 \pi b^3 + 12 \pi b^2 \][/tex]
The correct answer is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ 20 \pi b^3 + 12 \pi b^2 \][/tex]
