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Sagot :
To solve this problem, we'll calculate the volume and surface area of the tin using given formulas. The formulas we'll use are:
1. Volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
2. Surface area of a cylinder:
[tex]\[ A = 2\pi r h + 2\pi r^2 \][/tex]
which includes the lateral surface area and the area of the two circles (top and bottom).
### Given Values:
- Radius [tex]\( r = 4.2 \, \text{cm} \)[/tex]
- Height [tex]\( h = 8.4 \, \text{cm} \)[/tex]
### Step-by-Step Solution:
#### Volume Calculation:
1. Substitute [tex]\( r = 4.2 \, \text{cm} \)[/tex] and [tex]\( h = 8.4 \, \text{cm} \)[/tex] into the volume formula:
[tex]\[ V = \pi (4.2)^2 (8.4) \][/tex]
2. Perform the exponentiation and multiplication:
[tex]\[ V \approx 3.14159 \times 17.64 \times 8.4 \][/tex]
3. Calculate the final value:
[tex]\[ V \approx 465.5086 \, \text{cm}^3 \][/tex]
#### Surface Area Calculation:
1. Calculate the lateral surface area (side of the cylinder):
[tex]\[ A_{\text{lateral}} = 2 \pi r h \][/tex]
Substituting [tex]\( r = 4.2 \, \text{cm} \)[/tex] and [tex]\( h = 8.4 \, \text{cm} \)[/tex]:
[tex]\[ A_{\text{lateral}} = 2 \pi (4.2) (8.4) \][/tex]
2. Perform the multiplication:
[tex]\[ A_{\text{lateral}} \approx 2 \times 3.14159 \times 35.28 \][/tex]
[tex]\[ A_{\text{lateral}} \approx 221.6709 \, \text{cm}^2 \][/tex]
3. Calculate the top and bottom surface area (two circles):
[tex]\[ A_{\text{top and bottom}} = 2 \pi r^2 \][/tex]
Substituting [tex]\( r = 4.2 \, \text{cm} \)[/tex]:
[tex]\[ A_{\text{top and bottom}} = 2 \pi (4.2)^2 \][/tex]
4. Perform the multiplication:
[tex]\[ A_{\text{top and bottom}} \approx 2 \times 3.14159 \times 17.64 \][/tex]
[tex]\[ A_{\text{top and bottom}} \approx 110.8353 \, \text{cm}^2 \][/tex]
5. Add the lateral and top/bottom surface areas to get the total surface area:
[tex]\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{top and bottom}} \][/tex]
[tex]\[ A_{\text{total}} = 221.6709 + 110.8353 \][/tex]
[tex]\[ A_{\text{total}} \approx 332.5062 \, \text{cm}^2 \][/tex]
### Final Results:
- Volume [tex]\( V \approx 465.5086 \, \text{cm}^3 \)[/tex]
- Surface Area [tex]\( A \approx 332.5062 \, \text{cm}^2 \)[/tex]
### Comparison with Example:
When comparing this tin with the tin mentioned in the example that has a volume of 464 cm³, we notice the volumes are very close but not exactly the same. The tin in this problem has a slightly larger volume of approximately 465.5086 cm³, indicating minor differences possibly due to rounding or measurement precision.
1. Volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
2. Surface area of a cylinder:
[tex]\[ A = 2\pi r h + 2\pi r^2 \][/tex]
which includes the lateral surface area and the area of the two circles (top and bottom).
### Given Values:
- Radius [tex]\( r = 4.2 \, \text{cm} \)[/tex]
- Height [tex]\( h = 8.4 \, \text{cm} \)[/tex]
### Step-by-Step Solution:
#### Volume Calculation:
1. Substitute [tex]\( r = 4.2 \, \text{cm} \)[/tex] and [tex]\( h = 8.4 \, \text{cm} \)[/tex] into the volume formula:
[tex]\[ V = \pi (4.2)^2 (8.4) \][/tex]
2. Perform the exponentiation and multiplication:
[tex]\[ V \approx 3.14159 \times 17.64 \times 8.4 \][/tex]
3. Calculate the final value:
[tex]\[ V \approx 465.5086 \, \text{cm}^3 \][/tex]
#### Surface Area Calculation:
1. Calculate the lateral surface area (side of the cylinder):
[tex]\[ A_{\text{lateral}} = 2 \pi r h \][/tex]
Substituting [tex]\( r = 4.2 \, \text{cm} \)[/tex] and [tex]\( h = 8.4 \, \text{cm} \)[/tex]:
[tex]\[ A_{\text{lateral}} = 2 \pi (4.2) (8.4) \][/tex]
2. Perform the multiplication:
[tex]\[ A_{\text{lateral}} \approx 2 \times 3.14159 \times 35.28 \][/tex]
[tex]\[ A_{\text{lateral}} \approx 221.6709 \, \text{cm}^2 \][/tex]
3. Calculate the top and bottom surface area (two circles):
[tex]\[ A_{\text{top and bottom}} = 2 \pi r^2 \][/tex]
Substituting [tex]\( r = 4.2 \, \text{cm} \)[/tex]:
[tex]\[ A_{\text{top and bottom}} = 2 \pi (4.2)^2 \][/tex]
4. Perform the multiplication:
[tex]\[ A_{\text{top and bottom}} \approx 2 \times 3.14159 \times 17.64 \][/tex]
[tex]\[ A_{\text{top and bottom}} \approx 110.8353 \, \text{cm}^2 \][/tex]
5. Add the lateral and top/bottom surface areas to get the total surface area:
[tex]\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{top and bottom}} \][/tex]
[tex]\[ A_{\text{total}} = 221.6709 + 110.8353 \][/tex]
[tex]\[ A_{\text{total}} \approx 332.5062 \, \text{cm}^2 \][/tex]
### Final Results:
- Volume [tex]\( V \approx 465.5086 \, \text{cm}^3 \)[/tex]
- Surface Area [tex]\( A \approx 332.5062 \, \text{cm}^2 \)[/tex]
### Comparison with Example:
When comparing this tin with the tin mentioned in the example that has a volume of 464 cm³, we notice the volumes are very close but not exactly the same. The tin in this problem has a slightly larger volume of approximately 465.5086 cm³, indicating minor differences possibly due to rounding or measurement precision.
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