Question
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
=
2 in.
R
=
3 in.
h
=
19 in.
r
=
2 in.
R
=
3 in.
h
=
19 in.
Answers
GPT-4o mini
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:
\[
\boxed{298.30} \, \text{in}^3
\]
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:
\[
\boxed{298.30} \, \text{in}^3
\]