To find the volume of the Styrofoam collar, we can calculate the volume of the larger cylinder (with radius \( R \)) and subtract the volume of the smaller cylinder (with radius \( r \)). The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length) of the cylinder.
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Volume of the larger cylinder (radius \( R \)):
\[ V_{\text{large}} = \pi R^2 h = \pi (4 \text{ in.})^2 (16 \text{ in.}) \]
Calculating this:
\[ = \pi \cdot 16 \cdot 16 = 256 \pi \]
Now, using \( \pi \approx 3.14 \):
\[ V_{\text{large}} \approx 256 \cdot 3.14 = 804.64 \text{ in.}^3 \]
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Volume of the smaller cylinder (radius \( r \)):
\[ V_{\text{small}} = \pi r^2 h = \pi (2 \text{ in.})^2 (16 \text{ in.}) \]
Calculating this:
\[ = \pi \cdot 4 \cdot 16 = 64 \pi \]
Again using \( \pi \approx 3.14 \):
\[ V_{\text{small}} \approx 64 \cdot 3.14 = 200.96 \text{ in.}^3 \]
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Volume of the Styrofoam collar:
\[ V_{\text{collar}} = V_{\text{large}} - V_{\text{small}} = 804.64 \text{ in.}^3 - 200.96 \text{ in.}^3 \]
\[ V_{\text{collar}} \approx 603.68 \text{ in.}^3 \]
Finally, rounding to two decimal places, the volume of the Styrofoam collar is:
\[ \boxed{603.68} \text{ in.}^3 \]