To find the volume of the Styrofoam collar that insulates the pipe, we can consider it as a cylindrical shell. The volume \( V \) of the cylindrical shell can be calculated by finding the volume of the outer cylinder and subtracting the volume of the inner cylinder (the pipe).
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Step 1: Calculate the volume of the outer cylinder (radius \( R \))
Given:
- Outer radius \( R = 3 \) in.
- Height \( h = 19 \) in.
\[ V_{\text{outer}} = \pi R^2 h = \pi (3)^2 (19) \]
Calculating this:
\[ V_{\text{outer}} = \pi \cdot 9 \cdot 19 = \pi \cdot 171 \]
Substituting \( \pi \approx 3.14 \):
\[ V_{\text{outer}} \approx 3.14 \cdot 171 \approx 536.94 , \text{in}^3 \]
Step 2: Calculate the volume of the inner cylinder (radius \( r \))
Given:
- Inner radius \( r = 2 \) in.
- Height \( h = 19 \) in.
\[ V_{\text{inner}} = \pi r^2 h = \pi (2)^2 (19) \]
Calculating this:
\[ V_{\text{inner}} = \pi \cdot 4 \cdot 19 = \pi \cdot 76 \]
Substituting \( \pi \approx 3.14 \):
\[ V_{\text{inner}} \approx 3.14 \cdot 76 \approx 238.64 , \text{in}^3 \]
Step 3: Calculate the volume of the collar
Now we subtract the volume of the inner cylinder from the volume of the outer cylinder:
\[ V_{\text{collar}} = V_{\text{outer}} - V_{\text{inner}} \approx 536.94 - 238.64 \approx 298.30 , \text{in}^3 \]
Conclusion
The volume of the Styrofoam collar is approximately:
\[ \boxed{298.30} , \text{in}^3 \]
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