To find the volume of the cylindrical collar of Styrofoam that is insulating a pipe, we can calculate the volume of the larger cylinder (with radius \( R \)) and subtract the volume of the smaller cylinder (with radius \( r \)).
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Volume of the larger cylinder \( V_R \): \[ V_R = \pi R^2 h \] where:
- \( R = 6 \) inches
- \( h = 20 \) inches
Substituting in the values: \[ V_R = 3.14 \times (6)^2 \times 20 \] \[ V_R = 3.14 \times 36 \times 20 \] \[ V_R = 3.14 \times 720 \] \[ V_R = 2268 \text{ cubic inches} \]
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Volume of the smaller cylinder \( V_r \): \[ V_r = \pi r^2 h \] where:
- \( r = 3 \) inches
- \( h = 20 \) inches
Substituting in the values: \[ V_r = 3.14 \times (3)^2 \times 20 \] \[ V_r = 3.14 \times 9 \times 20 \] \[ V_r = 3.14 \times 180 \] \[ V_r = 565.2 \text{ cubic inches} \]
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Volume of the collar \( V_{collar} \): \[ V_{collar} = V_R - V_r \] \[ V_{collar} = 2268 - 565.2 \] \[ V_{collar} = 1702.8 \text{ cubic inches} \]
Thus, the volume of the collar of Styrofoam is 1702.8 cubic inches.