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Sagot :
To find the mass of the cylindrical iron pipe, we'll go through a series of calculations step-by-step. Here's the detailed solution:
1. Extract the Given Data:
- Length of the pipe (height, [tex]\( h \)[/tex]) = [tex]\( 3.5 \)[/tex] cm
- External diameter of the pipe = [tex]\( 18 \)[/tex] cm
- Thickness of the pipe ([tex]\( t \)[/tex]) = [tex]\( 1 \)[/tex] cm
- Density of the metal ([tex]\( \rho \)[/tex]) = [tex]\( 10.2 \)[/tex] g/cm[tex]\(^3\)[/tex]
- Use [tex]\(\pi = \frac{22}{7}\)[/tex]
2. Calculate the Radii:
- External radius ([tex]\( R \)[/tex]) is half of the external diameter:
[tex]\[ R = \frac{18}{2} = 9 \text{ cm} \][/tex]
- Internal radius ([tex]\( r \)[/tex]) is the external radius minus the thickness:
[tex]\[ r = R - t = 9 \text{ cm} - 1 \text{ cm} = 8 \text{ cm} \][/tex]
3. Calculate the Volumes:
- Volume of the external cylinder ([tex]\( V_{\text{external}} \)[/tex]):
[tex]\[ V_{\text{external}} = \pi R^2 h = \frac{22}{7} \times (9)^2 \times 3.5 \][/tex]
- Calculating the external volume:
[tex]\[ V_{\text{external}} = \frac{22}{7} \times 81 \times 3.5 \][/tex]
[tex]\[ V_{\text{external}} = \frac{22}{7} \times 283.5 = 22 \times 40.5 = 891 \text{ cm}^3 \][/tex]
- Volume of the internal cylinder ([tex]\( V_{\text{internal}} \)[/tex]):
[tex]\[ V_{\text{internal}} = \pi r^2 h = \frac{22}{7} \times (8)^2 \times 3.5 \][/tex]
- Calculating the internal volume:
[tex]\[ V_{\text{internal}} = \frac{22}{7} \times 64 \times 3.5 \][/tex]
[tex]\[ V_{\text{internal}} = \frac{22}{7} \times 224 = 22 \times 32 = 704 \text{ cm}^3 \][/tex]
- Volume of the metal (the material of the pipe) is the difference between the volumes of the external and internal cylinders:
[tex]\[ V_{\text{metal}} = V_{\text{external}} - V_{\text{internal}} = 891 \text{ cm}^3 - 704 \text{ cm}^3 = 187 \text{ cm}^3 \][/tex]
4. Calculate the Mass of the Pipe:
- The mass [tex]\( m \)[/tex] is given by the volume of the metal times the density:
[tex]\[ m = V_{\text{metal}} \times \rho = 187 \text{ cm}^3 \times 10.2 \text{ g/cm}^3 \][/tex]
- Calculating the mass:
[tex]\[ m = 187 \times 10.2 = 1907.4 \text{ g} \][/tex]
Hence, the external volume of the cylinder is [tex]\( 891 \text{ cm}^3 \)[/tex], the internal volume is [tex]\( 704 \text{ cm}^3 \)[/tex], the volume of the metal is [tex]\( 187 \text{ cm}^3 \)[/tex], and the mass of the cylindrical pipe is [tex]\( 1907.4 \text{ g} \)[/tex].
1. Extract the Given Data:
- Length of the pipe (height, [tex]\( h \)[/tex]) = [tex]\( 3.5 \)[/tex] cm
- External diameter of the pipe = [tex]\( 18 \)[/tex] cm
- Thickness of the pipe ([tex]\( t \)[/tex]) = [tex]\( 1 \)[/tex] cm
- Density of the metal ([tex]\( \rho \)[/tex]) = [tex]\( 10.2 \)[/tex] g/cm[tex]\(^3\)[/tex]
- Use [tex]\(\pi = \frac{22}{7}\)[/tex]
2. Calculate the Radii:
- External radius ([tex]\( R \)[/tex]) is half of the external diameter:
[tex]\[ R = \frac{18}{2} = 9 \text{ cm} \][/tex]
- Internal radius ([tex]\( r \)[/tex]) is the external radius minus the thickness:
[tex]\[ r = R - t = 9 \text{ cm} - 1 \text{ cm} = 8 \text{ cm} \][/tex]
3. Calculate the Volumes:
- Volume of the external cylinder ([tex]\( V_{\text{external}} \)[/tex]):
[tex]\[ V_{\text{external}} = \pi R^2 h = \frac{22}{7} \times (9)^2 \times 3.5 \][/tex]
- Calculating the external volume:
[tex]\[ V_{\text{external}} = \frac{22}{7} \times 81 \times 3.5 \][/tex]
[tex]\[ V_{\text{external}} = \frac{22}{7} \times 283.5 = 22 \times 40.5 = 891 \text{ cm}^3 \][/tex]
- Volume of the internal cylinder ([tex]\( V_{\text{internal}} \)[/tex]):
[tex]\[ V_{\text{internal}} = \pi r^2 h = \frac{22}{7} \times (8)^2 \times 3.5 \][/tex]
- Calculating the internal volume:
[tex]\[ V_{\text{internal}} = \frac{22}{7} \times 64 \times 3.5 \][/tex]
[tex]\[ V_{\text{internal}} = \frac{22}{7} \times 224 = 22 \times 32 = 704 \text{ cm}^3 \][/tex]
- Volume of the metal (the material of the pipe) is the difference between the volumes of the external and internal cylinders:
[tex]\[ V_{\text{metal}} = V_{\text{external}} - V_{\text{internal}} = 891 \text{ cm}^3 - 704 \text{ cm}^3 = 187 \text{ cm}^3 \][/tex]
4. Calculate the Mass of the Pipe:
- The mass [tex]\( m \)[/tex] is given by the volume of the metal times the density:
[tex]\[ m = V_{\text{metal}} \times \rho = 187 \text{ cm}^3 \times 10.2 \text{ g/cm}^3 \][/tex]
- Calculating the mass:
[tex]\[ m = 187 \times 10.2 = 1907.4 \text{ g} \][/tex]
Hence, the external volume of the cylinder is [tex]\( 891 \text{ cm}^3 \)[/tex], the internal volume is [tex]\( 704 \text{ cm}^3 \)[/tex], the volume of the metal is [tex]\( 187 \text{ cm}^3 \)[/tex], and the mass of the cylindrical pipe is [tex]\( 1907.4 \text{ g} \)[/tex].
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