Find the best solutions to your problems with the help of IDNLearn.com's experts. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
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].
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Find reliable answers at IDNLearn.com. Thanks for stopping by, and come back for more trustworthy solutions.
