To find the volume of the cylindrical can, we use the volume formula for a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
Given:
- Radius [tex]\( r = x + 8 \)[/tex]
- Height [tex]\( h = 2x + 3 \)[/tex]
First, we substitute these into the volume formula:
[tex]\[
V = \pi (x + 8)^2 (2x + 3)
\][/tex]
Now, let's expand the expression [tex]\( (x + 8)^2 \)[/tex]:
[tex]\[
(x + 8)^2 = (x + 8)(x + 8) = x^2 + 16x + 64
\][/tex]
Next, multiply this result by [tex]\( 2x + 3 \)[/tex]:
[tex]\[
\pi (x^2 + 16x + 64)(2x + 3)
\][/tex]
First, distribute [tex]\( 2x \)[/tex] across [tex]\( x^2 + 16x + 64 \)[/tex]:
[tex]\[
2x(x^2 + 16x + 64) = 2x^3 + 32x^2 + 128x
\][/tex]
Then, distribute [tex]\( 3 \)[/tex] across [tex]\( x^2 + 16x + 64 \)[/tex]:
[tex]\[
3(x^2 + 16x + 64) = 3x^2 + 48x + 192
\][/tex]
Combine the results by adding like terms:
[tex]\[
2x^3 + 32x^2 + 128x + 3x^2 + 48x + 192
\][/tex]
Simplify by combining like terms:
[tex]\[
2x^3 + (32x^2 + 3x^2) + (128x + 48x) + 192 = 2x^3 + 35x^2 + 176x + 192
\][/tex]
Thus, the expression for the volume of the can is:
[tex]\[
V = \pi (2x^3 + 35x^2 + 176x + 192)
\][/tex]
So the expression that represents the volume of the can is:
[tex]\[
2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi
\][/tex]
Therefore, the correct answer is:
[tex]\[
2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi
\][/tex]
From the given options, this matches:
[tex]\[
2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi
\][/tex]
