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 .

r
​=
5 in.
R
​=
8 in.
h
​=
17 in.

1 answer

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.