A collar of Styrofoam is made to insulate a pipe. Find its volume. The large radius R is to the outer rim. The small radius r is to the edge of the insulation. Use pi almost equals 3.14 .

Question

r
​=
3 in.
R
​=
6 in.
h
​=
20 in.

Answer 1

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 \( r \)).

1. Volume of the larger cylinder \( V_R \): \[ V_R = \pi R^2 h \] where:

   • \( R = 6 \) inches
   • \( h = 20 \) inches

   Substituting in the values: \[ V_R = 3.14 \times (6)^2 \times 20 \] \[ V_R = 3.14 \times 36 \times 20 \] \[ V_R = 3.14 \times 720 \] \[ V_R = 2268 \text{ cubic inches} \]

2. Volume of the smaller cylinder \( V_r \): \[ V_r = \pi r^2 h \] where:

   • \( r = 3 \) inches
   • \( h = 20 \) inches

   Substituting in the values: \[ V_r = 3.14 \times (3)^2 \times 20 \] \[ V_r = 3.14 \times 9 \times 20 \] \[ V_r = 3.14 \times 180 \] \[ V_r = 565.2 \text{ cubic inches} \]

3. 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.