To find the volume of the Styrofoam collar insulating the pipe, we need to calculate the volume of the larger cylinder (with outer radius \( R \) and height \( h \)) and subtract the volume of the smaller cylinder (with inner radius \( r \) and height \( h \)).
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Step 1: Calculate the volume of the larger cylinder
The larger cylinder has:
- Radius \( R = 8 , \text{in.} \)
- Height \( h = 17 , \text{in.} \)
Using the formula:
\[ V_{large} = \pi R^2 h = \pi (8)^2 (17) \]
Calculating \( (8)^2 \):
\[ (8)^2 = 64 \]
Now substituting back into the volume formula:
\[ V_{large} = \pi (64) (17) = 1088 \pi \]
Using \( \pi \approx 3.14 \):
\[ V_{large} \approx 1088 \times 3.14 = 3411.52 , \text{in}^3 \]
Step 2: Calculate the volume of the smaller cylinder
The smaller cylinder has:
- Radius \( r = 5 , \text{in.} \)
- Height \( h = 17 , \text{in.} \)
Using the volume formula:
\[ V_{small} = \pi r^2 h = \pi (5)^2 (17) \]
Calculating \( (5)^2 \):
\[ (5)^2 = 25 \]
Substituting back into the volume formula:
\[ V_{small} = \pi (25) (17) = 425 \pi \]
Using \( \pi \approx 3.14 \):
\[ V_{small} \approx 425 \times 3.14 = 1333.5 , \text{in}^3 \]
Step 3: Find the volume of the collar
Now we find the volume of the collar by subtracting the volume of the smaller cylinder from the volume of the larger cylinder:
\[ V_{collar} = V_{large} - V_{small} \]
Calculating:
\[ V_{collar} \approx 3411.52 - 1333.5 = 2078.02 , \text{in}^3 \]
Conclusion
The volume of the Styrofoam collar is approximately 2078.02 cubic inches.