To determine which expression represents the volume of the cylindrical can, let's break down the steps:
1. Identify the given components for radius and height:
- The radius [tex]\( r \)[/tex] is given by [tex]\( r = x + 8 \)[/tex].
- The height [tex]\( h \)[/tex] is given by [tex]\( h = 2x + 3 \)[/tex].
2. Write the formula for the volume [tex]\( V \)[/tex] of a cylinder:
The volume [tex]\( V \)[/tex] is given by:
[tex]\[
V = \pi r^2 h
\][/tex]
3. Substitute the given expressions for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[
V = \pi (x + 8)^2 (2x + 3)
\][/tex]
4. Expand [tex]\( (x + 8)^2 \)[/tex]:
[tex]\[
(x + 8)^2 = x^2 + 16x + 64
\][/tex]
5. Substitute this expansion into the volume formula:
[tex]\[
V = \pi (x^2 + 16x + 64)(2x + 3)
\][/tex]
6. Expand the product [tex]\((x^2 + 16x + 64)(2x + 3)\)[/tex]:
[tex]\[
(x^2 + 16x + 64)(2x + 3) = x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3)
\][/tex]
[tex]\[
= 2x^3 + 3x^2 + 32x^2 + 48x + 128x + 192
\][/tex]
[tex]\[
= 2x^3 + 35x^2 + 176x + 192
\][/tex]
7. Multiply the expanded polynomial by [tex]\( \pi \)[/tex]:
[tex]\[
V = \pi (2x^3 + 35x^2 + 176x + 192)
\][/tex]
8. Compare the expanded volume expression with the given options:
[tex]\[
\pi x^3 + 19\pi x^2 + 112\pi x + 192\pi
\][/tex]
[tex]\[
2\pi x^3 + 35\pi x^2 + 80\pi x + 48\pi
\][/tex]
[tex]\[
2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi
\][/tex]
[tex]\[
4\pi x^3 + 44\pi x^2 + 105\pi x + 72\pi
\][/tex]
The expression that matches our expanded result is:
[tex]\[
2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi
\][/tex]
So the correct expression representing the volume of the can is:
[tex]\[ \boxed{2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi} \][/tex]
