To determine which argument is correct, let's follow the detailed mathematical steps to find the base area and volume of cone [tex]\( W \)[/tex] and compare them with the volume of square pyramid [tex]\( X \)[/tex].
### Step-by-Step Solution:
1. Base Area of Cone [tex]\( W \)[/tex]:
- The formula for the base area of a cone is given by [tex]\( \pi r^2 \)[/tex].
- Given [tex]\( r = 10 \)[/tex] cm, the base area is:
[tex]\[
\pi (10^2) = 314.1592653589793 \, \text{cm}^2
\][/tex]
2. Volume of Cone [tex]\( W \)[/tex]:
- The formula for the volume of a cone is [tex]\( \frac{1}{3} \text{(base area)} \times \text{height} \)[/tex].
- Given the base area [tex]\( = 314.1592653589793 \, \text{cm}^2 \)[/tex] and height [tex]\( h = 5 \)[/tex] cm, the volume is:
[tex]\[
\frac{1}{3} (314.1592653589793) \times 5 = 523.5987755982989 \, \text{cm}^3
\][/tex]
3. Volume of Square Pyramid [tex]\( X \)[/tex]:
- Square pyramid [tex]\( X \)[/tex] has the same base area and height as cone [tex]\( W \)[/tex]. The formula for the volume of a square pyramid is also [tex]\( \frac{1}{3} \text{(base area)} \times \text{height} \)[/tex].
- Given the base area [tex]\( = 314.1592653589793 \, \text{cm}^2 \)[/tex] and height [tex]\( h = 5 \)[/tex] cm, the volume is:
[tex]\[
\frac{1}{3} (314.1592653589793) \times 5 = 523.5987755982989 \, \text{cm}^3
\][/tex]
From the above calculations, we see that both the volume of cone [tex]\( W \)[/tex] and the volume of square pyramid [tex]\( X \)[/tex] are equal, each being [tex]\( 523.5987755982989 \, \text{cm}^3 \)[/tex].
Let's examine both arguments:
- Paul's Argument:
- Paul asserts that the volume of square pyramid [tex]\( X \)[/tex] is three times the volume of cone [tex]\( W \)[/tex].
- According to Paul's calculation, the volume of cone [tex]\( W \)[/tex] is correctly [tex]\( 523.33 \, \text{cm}^3 \)[/tex], but he incorrectly finds the volume of pyramid [tex]\( X \)[/tex] as [tex]\( 1,570 \, \text{cm}^3 \)[/tex] using the formula [tex]\( \text{base area} \times \text{height} \)[/tex], which is wrong.
- Manuel's Argument:
- Manuel asserts that the volume of square pyramid [tex]\( X \)[/tex] is equal to the volume of cone [tex]\( W \)[/tex].
- Manuel correctly calculates both volumes as [tex]\( 523.33 \, \text{cm}^3 \)[/tex] (note: the slight difference in rounding here).
Based on the given calculations and discussion, Manuel's argument is correct. Paul used an incorrect formula for the volume of the square pyramid.
Correct statement: Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid [tex]\( X \)[/tex].
