Answer :
To find the mass of a cylindrical iron pipe with the given dimensions and density, we need to go through the following detailed steps:
1. Given Values:
- Length (height) of the pipe, [tex]\(h = 3.5 \, \text{cm}\)[/tex]
- External diameter of the pipe, [tex]\(D_\text{ext} = 18 \, \text{cm}\)[/tex]
- Thickness of the pipe, [tex]\(t = 1 \, \text{cm}\)[/tex]
- Density of the material, [tex]\(\rho = 10.2 \, \text{g/cm}^3\)[/tex]
- Value of [tex]\(\pi = \frac{22}{7}\)[/tex]
2. Calculate the Internal Diameter:
The internal diameter can be found by subtracting twice the thickness from the external diameter:
[tex]\[ D_\text{int} = D_\text{ext} - 2 \cdot t \][/tex]
So,
[tex]\[ D_\text{int} = 18 \, \text{cm} - 2 \cdot 1 \, \text{cm} = 16 \, \text{cm} \][/tex]
3. Calculate the Radii:
- External radius, [tex]\(R_\text{ext}\)[/tex]:
[tex]\[ R_\text{ext} = \frac{D_\text{ext}}{2} = \frac{18 \, \text{cm}}{2} = 9 \, \text{cm} \][/tex]
- Internal radius, [tex]\(R_\text{int}\)[/tex]:
[tex]\[ R_\text{int} = \frac{D_\text{int}}{2} = \frac{16 \, \text{cm}}{2} = 8 \, \text{cm} \][/tex]
4. Calculate the External Volume of the Cylinder:
The volume of a cylinder is given by [tex]\(\pi r^2 h\)[/tex]. For the external volume:
[tex]\[ V_\text{ext} = \pi R_\text{ext}^2 h \][/tex]
Substitute the values:
[tex]\[ V_\text{ext} = \frac{22}{7} \cdot (9 \, \text{cm})^2 \cdot 3.5 \, \text{cm} \][/tex]
This results in:
[tex]\[ V_\text{ext} = 891.0 \, \text{cm}^3 \][/tex]
5. Calculate the Internal Volume of the Cylinder:
Similarly, the internal volume is:
[tex]\[ V_\text{int} = \pi R_\text{int}^2 h \][/tex]
Substitute the values:
[tex]\[ V_\text{int} = \frac{22}{7} \cdot (8 \, \text{cm})^2 \cdot 3.5 \, \text{cm} \][/tex]
This results in:
[tex]\[ V_\text{int} = 704.0 \, \text{cm}^3 \][/tex]
6. Calculate the Volume of the Metal:
The volume of the metal part of the pipe is the difference between the external and internal volumes:
[tex]\[ V_\text{metal} = V_\text{ext} - V_\text{int} \][/tex]
So,
[tex]\[ V_\text{metal} = 891.0 \, \text{cm}^3 - 704.0 \, \text{cm}^3 = 187.0 \, \text{cm}^3 \][/tex]
7. Calculate the Mass of the Cylindrical Pipe:
The mass can be calculated by multiplying the volume of the metal by the density:
[tex]\[ \text{Mass} = V_\text{metal} \cdot \rho \][/tex]
Substitute the values:
[tex]\[ \text{Mass} = 187.0 \, \text{cm}^3 \cdot 10.2 \, \text{g/cm}^3 \][/tex]
This results in:
[tex]\[ \text{Mass} = 1907.4 \, \text{g} \][/tex]
Therefore, the mass of the cylindrical iron pipe is approximately [tex]\(1907.4 \, \text{g}\)[/tex].
1. Given Values:
- Length (height) of the pipe, [tex]\(h = 3.5 \, \text{cm}\)[/tex]
- External diameter of the pipe, [tex]\(D_\text{ext} = 18 \, \text{cm}\)[/tex]
- Thickness of the pipe, [tex]\(t = 1 \, \text{cm}\)[/tex]
- Density of the material, [tex]\(\rho = 10.2 \, \text{g/cm}^3\)[/tex]
- Value of [tex]\(\pi = \frac{22}{7}\)[/tex]
2. Calculate the Internal Diameter:
The internal diameter can be found by subtracting twice the thickness from the external diameter:
[tex]\[ D_\text{int} = D_\text{ext} - 2 \cdot t \][/tex]
So,
[tex]\[ D_\text{int} = 18 \, \text{cm} - 2 \cdot 1 \, \text{cm} = 16 \, \text{cm} \][/tex]
3. Calculate the Radii:
- External radius, [tex]\(R_\text{ext}\)[/tex]:
[tex]\[ R_\text{ext} = \frac{D_\text{ext}}{2} = \frac{18 \, \text{cm}}{2} = 9 \, \text{cm} \][/tex]
- Internal radius, [tex]\(R_\text{int}\)[/tex]:
[tex]\[ R_\text{int} = \frac{D_\text{int}}{2} = \frac{16 \, \text{cm}}{2} = 8 \, \text{cm} \][/tex]
4. Calculate the External Volume of the Cylinder:
The volume of a cylinder is given by [tex]\(\pi r^2 h\)[/tex]. For the external volume:
[tex]\[ V_\text{ext} = \pi R_\text{ext}^2 h \][/tex]
Substitute the values:
[tex]\[ V_\text{ext} = \frac{22}{7} \cdot (9 \, \text{cm})^2 \cdot 3.5 \, \text{cm} \][/tex]
This results in:
[tex]\[ V_\text{ext} = 891.0 \, \text{cm}^3 \][/tex]
5. Calculate the Internal Volume of the Cylinder:
Similarly, the internal volume is:
[tex]\[ V_\text{int} = \pi R_\text{int}^2 h \][/tex]
Substitute the values:
[tex]\[ V_\text{int} = \frac{22}{7} \cdot (8 \, \text{cm})^2 \cdot 3.5 \, \text{cm} \][/tex]
This results in:
[tex]\[ V_\text{int} = 704.0 \, \text{cm}^3 \][/tex]
6. Calculate the Volume of the Metal:
The volume of the metal part of the pipe is the difference between the external and internal volumes:
[tex]\[ V_\text{metal} = V_\text{ext} - V_\text{int} \][/tex]
So,
[tex]\[ V_\text{metal} = 891.0 \, \text{cm}^3 - 704.0 \, \text{cm}^3 = 187.0 \, \text{cm}^3 \][/tex]
7. Calculate the Mass of the Cylindrical Pipe:
The mass can be calculated by multiplying the volume of the metal by the density:
[tex]\[ \text{Mass} = V_\text{metal} \cdot \rho \][/tex]
Substitute the values:
[tex]\[ \text{Mass} = 187.0 \, \text{cm}^3 \cdot 10.2 \, \text{g/cm}^3 \][/tex]
This results in:
[tex]\[ \text{Mass} = 1907.4 \, \text{g} \][/tex]
Therefore, the mass of the cylindrical iron pipe is approximately [tex]\(1907.4 \, \text{g}\)[/tex].
