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Sagot :
To determine the volume of metal needed to make the pipe, we need to follow these steps:
1. Calculate the volume of the entire cylindrical pipe:
- Diameter of the pipe [tex]\( D = 20 \)[/tex] mm.
- Radius of the pipe [tex]\( r_{\text{pipe}} = \frac{D}{2} = \frac{20}{2} = 10 \)[/tex] mm.
- Height of the pipe [tex]\( h = 21 \)[/tex] mm.
- Volume of the pipe [tex]\( V_{\text{pipe}} \)[/tex] is given by the formula for the volume of a cylinder: [tex]\( V = \pi r^2 h \)[/tex].
[tex]\[ V_{\text{pipe}} = \pi (10)^2 (21) = \pi \times 100 \times 21 = 2100 \pi \text{ cubic millimeters} \][/tex]
2. Calculate the volume of the cylindrical hole:
- Radius of the hole [tex]\( r_{\text{hole}} = 6 \)[/tex] mm.
- Volume of the hole [tex]\( V_{\text{hole}} \)[/tex] is given by the same formula for the volume of a cylinder: [tex]\( V = \pi r^2 h \)[/tex].
[tex]\[ V_{\text{hole}} = \pi (6)^2 (21) = \pi \times 36 \times 21 = 756 \pi \text{ cubic millimeters} \][/tex]
3. Calculate the volume of the metal needed:
- This is the volume of the pipe minus the volume of the hole.
[tex]\[ V_{\text{metal}} = V_{\text{pipe}} - V_{\text{hole}} = 2100 \pi - 756 \pi = 1344 \pi \text{ cubic millimeters} \][/tex]
Given the detailed steps above, let’s check the provided expressions against this calculated volume:
1. [tex]\( 21 \pi(10)^2-21 \pi(6)^2 \)[/tex]
[tex]\[ 21 \pi (10)^2 - 21 \pi (6)^2 = 21 \pi (100) - 21 \pi (36) = 2100 \pi - 756 \pi = 1344 \pi \][/tex]
This expression is correct.
2. [tex]\( \pi(20)^2(21)-\pi(6)^2 \)[/tex]
[tex]\[ \pi (20)^2 (21) - \pi (6)^2 = \pi (400) (21) - \pi (36) = 8400 \pi - 36 \pi \][/tex]
This expression is incorrect.
3. [tex]\( 2100 \pi - 756 \pi \)[/tex]
[tex]\[ 2100 \pi - 756 \pi = 1344 \pi \][/tex]
This expression is correct.
4. [tex]\( 7,644 \pi \)[/tex]
[tex]\[ 7644 \pi \,\text{cubic millimeters} \][/tex]
This expression is incorrect as it does not match the calculated volume.
5. [tex]\( 1,344 \)[/tex]
[tex]\[ 1344 \,\text{cubic millimeters} \][/tex]
This expression is incorrect as it is not in terms of [tex]\(\pi\)[/tex].
So, the correct expressions representing the volume of metal needed to make the pipe are:
1. [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
3. [tex]\( 2100 \pi - 756 \pi \)[/tex]
1. Calculate the volume of the entire cylindrical pipe:
- Diameter of the pipe [tex]\( D = 20 \)[/tex] mm.
- Radius of the pipe [tex]\( r_{\text{pipe}} = \frac{D}{2} = \frac{20}{2} = 10 \)[/tex] mm.
- Height of the pipe [tex]\( h = 21 \)[/tex] mm.
- Volume of the pipe [tex]\( V_{\text{pipe}} \)[/tex] is given by the formula for the volume of a cylinder: [tex]\( V = \pi r^2 h \)[/tex].
[tex]\[ V_{\text{pipe}} = \pi (10)^2 (21) = \pi \times 100 \times 21 = 2100 \pi \text{ cubic millimeters} \][/tex]
2. Calculate the volume of the cylindrical hole:
- Radius of the hole [tex]\( r_{\text{hole}} = 6 \)[/tex] mm.
- Volume of the hole [tex]\( V_{\text{hole}} \)[/tex] is given by the same formula for the volume of a cylinder: [tex]\( V = \pi r^2 h \)[/tex].
[tex]\[ V_{\text{hole}} = \pi (6)^2 (21) = \pi \times 36 \times 21 = 756 \pi \text{ cubic millimeters} \][/tex]
3. Calculate the volume of the metal needed:
- This is the volume of the pipe minus the volume of the hole.
[tex]\[ V_{\text{metal}} = V_{\text{pipe}} - V_{\text{hole}} = 2100 \pi - 756 \pi = 1344 \pi \text{ cubic millimeters} \][/tex]
Given the detailed steps above, let’s check the provided expressions against this calculated volume:
1. [tex]\( 21 \pi(10)^2-21 \pi(6)^2 \)[/tex]
[tex]\[ 21 \pi (10)^2 - 21 \pi (6)^2 = 21 \pi (100) - 21 \pi (36) = 2100 \pi - 756 \pi = 1344 \pi \][/tex]
This expression is correct.
2. [tex]\( \pi(20)^2(21)-\pi(6)^2 \)[/tex]
[tex]\[ \pi (20)^2 (21) - \pi (6)^2 = \pi (400) (21) - \pi (36) = 8400 \pi - 36 \pi \][/tex]
This expression is incorrect.
3. [tex]\( 2100 \pi - 756 \pi \)[/tex]
[tex]\[ 2100 \pi - 756 \pi = 1344 \pi \][/tex]
This expression is correct.
4. [tex]\( 7,644 \pi \)[/tex]
[tex]\[ 7644 \pi \,\text{cubic millimeters} \][/tex]
This expression is incorrect as it does not match the calculated volume.
5. [tex]\( 1,344 \)[/tex]
[tex]\[ 1344 \,\text{cubic millimeters} \][/tex]
This expression is incorrect as it is not in terms of [tex]\(\pi\)[/tex].
So, the correct expressions representing the volume of metal needed to make the pipe are:
1. [tex]\( 21 \pi (10)^2 - 21 \pi (6)^2 \)[/tex]
3. [tex]\( 2100 \pi - 756 \pi \)[/tex]
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