To determine the relationship between the volume of the given cylinder and cone, we need to follow a series of steps. This includes calculating the volume of each shape and then comparing them.
### Step 1: Calculate the Volume of the Cylinder
Given:
- Diameter of the cylinder, [tex]\( d \)[/tex] = 8 inches
- Radius of the cylinder, [tex]\( r \)[/tex] = [tex]\( d/2 \)[/tex] = 8/2 = 4 inches
- Height of the cylinder, [tex]\( h \)[/tex] = 3 inches
- Value of π = 3.14
The formula for the volume of a cylinder is:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Plugging in the values:
[tex]\[ V_{\text{cylinder}} = 3.14 \times (4)^2 \times 3 \][/tex]
[tex]\[ V_{\text{cylinder}} = 3.14 \times 16 \times 3 \][/tex]
[tex]\[ V_{\text{cylinder}} = 3.14 \times 48 \][/tex]
[tex]\[ V_{\text{cylinder}} = 150.72 \, \text{cubic inches} \][/tex]
### Step 2: Calculate the Volume of the Cone
Given:
- Diameter of the cone, [tex]\( d \)[/tex] = 8 inches
- Radius of the cone, [tex]\( r \)[/tex] = [tex]\( d/2 \)[/tex] = 8/2 = 4 inches
- Height of the cone, [tex]\( h \)[/tex] = 18 inches
- Value of π = 3.14
The formula for the volume of a cone is:
[tex]\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 h \][/tex]
Plugging in the values:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \times 3.14 \times (4)^2 \times 18 \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 16 \times 18 \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \times 3.14 \times 288 \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \times 904.32 \][/tex]
[tex]\[ V_{\text{cone}} = 301.44 \, \text{cubic inches} \][/tex]
### Step 3: Compare the Volumes
Now that we have the volumes of both the cylinder and the cone, we can compare them:
- Volume of the cylinder, [tex]\( V_{\text{cylinder}} \)[/tex] = 150.72 cubic inches
- Volume of the cone, [tex]\( V_{\text{cone}} \)[/tex] = 301.44 cubic inches
To find the relationship between the two volumes, we can calculate the ratio of the cylinder's volume to the cone's volume:
[tex]\[ \text{Volume Ratio} = \frac{V_{\text{cylinder}}}{V_{\text{cone}}} \][/tex]
Plugging in the volumes:
[tex]\[ \text{Volume Ratio} = \frac{150.72}{301.44} \][/tex]
[tex]\[ \text{Volume Ratio} = 0.5 \][/tex]
### Conclusion
The volume of the cylinder is half the volume of the cone.
Therefore, the relationship between the volume of the cylinder and the cone is that the cylinder's volume is exactly 50% of the cone's volume.
