To determine which numerical expression gives the volume of the box that is filled with packing material, we need to start by calculating the volumes of both the cube and the cylinder.
1. Volume of the cube:
The cube has an interior side length of 18 inches. The formula for the volume of a cube is given by [tex]\( V_{\text{cube}} = \text{side}^3 \)[/tex].
So,
[tex]\[
V_{\text{cube}} = 18^3 = 5832 \, \text{cubic inches}.
\][/tex]
2. Volume of the cylinder:
The cylinder has a radius of 6 inches and a height of 12 inches. The formula for the volume of a cylinder is given by [tex]\( V_{\text{cylinder}} = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
So,
[tex]\[
V_{\text{cylinder}} = \pi(6^2)(12) = \pi (36)(12) = 432\pi \approx 1357.17 \, \text{cubic inches}.
\][/tex]
3. Volume filled with packing material:
To find the volume filled with packing material, we subtract the volume of the cylinder from the volume of the cube:
[tex]\[
V_{\text{packing material}} = V_{\text{cube}} - V_{\text{cylinder}} = 5832 - 1357.17 \approx 4474.83 \, \text{cubic inches}.
\][/tex]
Next, we compare the given numerical expressions to determine which one correctly represents the volume filled with the packing material:
- Option (A): [tex]\( 18^3 - \pi (12)^3 \)[/tex]
Calculating this,
[tex]\[
5832 - \pi (12^3) = 5832 - \pi (1728) \approx 5832 - 5428.67 \approx 403.33.
\][/tex]
- Option (B): [tex]\( 18^3 - \pi (6)^2 (12) \)[/tex]
Calculating this,
[tex]\[
5832 - \pi (6^2) (12) = 5832 - \pi (36) (12) = 5832 - 432\pi \approx 5832 - 1357.17 \approx 4474.83.
\][/tex]
- Option (C): [tex]\( 18^3 - \pi (6)(12)^2 \)[/tex]
Calculating this,
[tex]\[
5832 - \pi (6) (12^2) = 5832 - \pi (6) (144) = 5832 - 864\pi \approx 5832 - 2714.34 \approx 3117.66.
\][/tex]
- Option (D): [tex]\( 6 (18)^2 - 2\pi (6)(12) \)[/tex]
Calculating this,
[tex]\[
6 (18^2) - 2\pi (6)(12) = 6(324) - 2 \pi (72) = 1944 - 144\pi \approx 1944 - 452.39 \approx 1491.61.
\][/tex]
- Option (E): [tex]\( 6 (18)^2 - 2\pi (6)(12) - 2\pi (6)^2 \)[/tex]
Calculating this,
[tex]\[
6(18^2) - 2\pi (6)(12) - 2\pi (6)^2 = 6(324) - 2 \pi (72) - 2 \pi (36) = 1944 - 144\pi - 72\pi = 1944 - 216\pi \approx 1944 - 678.58 \approx 1265.42.
\][/tex]
After comparing all the options, the correct numerical expression that represents the volume filled with the packing material is:
[tex]\[
\boxed{18^3 - \pi (6)^2 (12)}
\][/tex]
Therefore, the correct answer is (B).
