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Sagot :
To solve this problem, let's analyze the calculations and arguments made by both Paul and Manuel regarding the volumes of cone [tex]\( W \)[/tex] and square pyramid [tex]\( X \)[/tex].
### Step-by-Step Solution:
1. Base Area Calculation:
Both objects, cone [tex]\( W \)[/tex] and square pyramid [tex]\( X \)[/tex], have the same base area.
The base area of cone [tex]\( W \)[/tex], which is also the base area of square pyramid [tex]\( X \)[/tex], is given by:
[tex]\[ \text{Base Area} = \pi r^2 = \pi (8^2) = \pi \cdot 64 = 201.06 \, \text{cm}^2 \][/tex]
2. Volume of Cone [tex]\( W \)[/tex]:
The volume of cone [tex]\( W \)[/tex] is calculated using the formula for the volume of a cone:
[tex]\[ \text{Volume of cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
3. Volume of Square Pyramid [tex]\( X \)[/tex]:
Both Paul and Manuel are calculating the volume of square pyramid [tex]\( X \)[/tex], but let's evaluate the correct approach.
The correct formula for the volume of a square pyramid is:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \text{Base Area} \times \text{Height} \][/tex]
Applying the given values:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
### Analyzing the Arguments:
- Paul's Argument:
Paul correctly uses the formula [tex]\(\frac{1}{3} \text{Base Area} \times \text{Height}\)[/tex] to calculate the volume of the square pyramid [tex]\( X \)[/tex]. His calculation:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
matches the volume of the cone [tex]\( W \)[/tex].
- Manuel's Argument:
Manuel incorrectly multiplies the base area by the height without the [tex]\(\frac{1}{3}\)[/tex] factor:
[tex]\[ \text{Volume of pyramid} = 201.06 \cdot 5 = 1005.30 \, \text{cm}^3 \][/tex]
This result is incorrect because he did not use the correct formula for the volume of a pyramid.
### Conclusion:
Based on the correct application of geometric volume formulas:
- Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid [tex]\( X \)[/tex].
Thus, the correct statement is:
Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid [tex]\( X \)[/tex].
### Step-by-Step Solution:
1. Base Area Calculation:
Both objects, cone [tex]\( W \)[/tex] and square pyramid [tex]\( X \)[/tex], have the same base area.
The base area of cone [tex]\( W \)[/tex], which is also the base area of square pyramid [tex]\( X \)[/tex], is given by:
[tex]\[ \text{Base Area} = \pi r^2 = \pi (8^2) = \pi \cdot 64 = 201.06 \, \text{cm}^2 \][/tex]
2. Volume of Cone [tex]\( W \)[/tex]:
The volume of cone [tex]\( W \)[/tex] is calculated using the formula for the volume of a cone:
[tex]\[ \text{Volume of cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
3. Volume of Square Pyramid [tex]\( X \)[/tex]:
Both Paul and Manuel are calculating the volume of square pyramid [tex]\( X \)[/tex], but let's evaluate the correct approach.
The correct formula for the volume of a square pyramid is:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \text{Base Area} \times \text{Height} \][/tex]
Applying the given values:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
### Analyzing the Arguments:
- Paul's Argument:
Paul correctly uses the formula [tex]\(\frac{1}{3} \text{Base Area} \times \text{Height}\)[/tex] to calculate the volume of the square pyramid [tex]\( X \)[/tex]. His calculation:
[tex]\[ \text{Volume of pyramid} = \frac{1}{3} \cdot 201.06 \cdot 5 = 335.10 \, \text{cm}^3 \][/tex]
matches the volume of the cone [tex]\( W \)[/tex].
- Manuel's Argument:
Manuel incorrectly multiplies the base area by the height without the [tex]\(\frac{1}{3}\)[/tex] factor:
[tex]\[ \text{Volume of pyramid} = 201.06 \cdot 5 = 1005.30 \, \text{cm}^3 \][/tex]
This result is incorrect because he did not use the correct formula for the volume of a pyramid.
### Conclusion:
Based on the correct application of geometric volume formulas:
- Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid [tex]\( X \)[/tex].
Thus, the correct statement is:
Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid [tex]\( X \)[/tex].
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