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10. (Multiple Choice Worth 1 point)

Cone [tex]$W$[/tex] has a radius of 10 cm and a height of 5 cm. Square pyramid [tex]$X$[/tex] has the same base area and height as cone [tex]$W$[/tex].

Paul and Manuel disagree on how the volumes of cone [tex]$W$[/tex] and square pyramid [tex]$X$[/tex] are related. Examine their arguments. Which statement explains whose argument is correct and why?

\begin{tabular}{|c|c|}
\hline
Paul & Manuel \\
\hline
\begin{tabular}{l}
The volume of square pyramid [tex]$X$[/tex] is three times the volume of cone [tex]$W$[/tex]. This can \\
be proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume \\
of square pyramid [tex]$X$[/tex]. The base area of cone [tex]$W$[/tex] is [tex]$\pi(r^2) = \pi(10^2) = 314 \, \text{cm}^2$[/tex]. The \\
volume of cone [tex]$W$[/tex] is [tex]$\frac{1}{3}$[/tex] (area of base) [tex]$(h) = \frac{1}{3}(314)(5) = 523.33 \, \text{cm}^3$[/tex]. The volume \\
of square pyramid [tex]$X$[/tex] is (area of base) [tex]$(h) = (314)(5) = 1,570 \, \text{cm}^3$[/tex].
\end{tabular}
&
\begin{tabular}{l}
The volume of square pyramid [tex]$X$[/tex] is equal to the volume of cone [tex]$W$[/tex]. This can be \\
proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume of \\
square pyramid [tex]$X$[/tex]. The base area of cone [tex]$W$[/tex] is [tex]$\pi(r^2) = \pi(10^2) = 314 \, \text{cm}^2$[/tex]. The \\
volume of cone [tex]$W$[/tex] is [tex]$\frac{1}{3}$[/tex] (area of base) [tex]$(h) = \frac{1}{3}(314)(5) = 523.33 \, \text{cm}^3$[/tex]. The volume \\
of square pyramid [tex]$X$[/tex] is [tex]$\frac{1}{3}$[/tex] (area of base) [tex]$(h) = \frac{1}{3}(314)(5) = 523.33 \, \text{cm}^3$[/tex].
\end{tabular} \\
\hline
\end{tabular}

A. Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid [tex]$X$[/tex].
B. Paul's argument is correct; Manuel used the incorrect base area to find the volume of square pyramid [tex]$X$[/tex].
C. Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid [tex]$X$[/tex].
D. Manuel's argument is correct; Paul used the incorrect base area to find the volume of square pyramid [tex]$X$[/tex].



Answer :

GinnyAnswer
Let's break down and analyze each person's argument to determine the correct statement:

### Paul's Argument:
- Paul states that the volume of square pyramid [tex]\(X\)[/tex] is three times the volume of cone [tex]\(W\)[/tex].
- He calculates the base area of the cone [tex]\(W\)[/tex] as [tex]\(\pi \times (10^2) = 314 \, \text{cm}^2\)[/tex].
- The volume of cone [tex]\(W\)[/tex] is calculated as [tex]\(\frac{1}{3} \times 314 \times 5 = 523.33 \, \text{cm}^3\)[/tex].
- Paul then claims the volume of square pyramid [tex]\(X\)[/tex] is simply the base area times the height: [tex]\(314 \times 5 = 1570 \, \text{cm}^3\)[/tex], concluding that it's three times the volume of the cone.

### Manuel's Argument:
- Manuel states that the volume of square pyramid [tex]\(X\)[/tex] is equal to the volume of cone [tex]\(W\)[/tex].
- He also calculates the base area of the cone [tex]\(W\)[/tex] as [tex]\(\pi \times (10^2) = 314 \, \text{cm}^2\)[/tex].
- The volume of cone [tex]\(W\)[/tex] he calculates is [tex]\(\frac{1}{3} \times 314 \times 5 = 523.33 \, \text{cm}^3\)[/tex].
- Manuel calculates the volume of square pyramid [tex]\(X\)[/tex] using the correct formula for the volume of a pyramid: [tex]\(\frac{1}{3} \times \text{base area} \times \text{height}\)[/tex].
- Therefore, the volume of square pyramid [tex]\(X\)[/tex] is also [tex]\(\frac{1}{3} \times 314 \times 5 = 523.33 \, \text{cm}^3\)[/tex].

### Comparing Both Arguments:
- Paul's calculation of the volume of the pyramid does not use the correct formula. He overlooked the [tex]\(\frac{1}{3}\)[/tex] factor essential for calculating the volume of a pyramid.
- Manuel used the correct formula for both the cone and the pyramid, leading to the same volume for both shapes.

Thus, Manuel's argument is correct because he applied the correct volume formula for the pyramid, resulting in the correct volume calculation showing the two volumes are equal.

### Statement that explains why:
- "Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid [tex]\(X\)[/tex]."