Cone W has a radius of 6 cm and a height of 5 cm. Square pyramid X has the same base area and height as cone W.
Paul and Manuel disagree on the reason why the volumes of cone W and square pyramid X are related.
| Paul | Manuel |
|---|---|
| The volume of square pyramid X is equal to the volume of cone W. This can be proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume of square pyramid X. | The volume of square pyramid X is equal to the volume of cone W. This can be proven by finding the base area and volume of cone [tex]$W$[/tex], along with the volume of square pyramid X. |
| The base area of cone [tex]$W$[/tex] is [tex]$n(d)=n(12)=37.68 cm^2$[/tex]. | The base area of cone [tex]$W$[/tex] is [tex]$\pi(r^2)=\pi(6^2)=113.04 cm^2$[/tex]. |
| The volume of cone W is [tex]$\frac{1}{3}[/tex] (area of base)( h )=[tex]$\frac{1}{3}(37.68)(5)=62.8$[/tex] [tex]$cm ^3$[/tex]. | The volume of cone [tex]$W$[/tex] is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}(113.04)(5)=[/tex] [tex]$188.4 cm^3$[/tex]. |
| The volume of square pyramid X is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}(37.68)$[/tex] (5) [tex]$=62.8 cm^3$[/tex]. | The volume of square pyramid [tex]$X$[/tex] is [tex]$\frac{1}{3}[/tex] (area of base)(h) =[tex]$\frac{1}{3}$[/tex] [tex]$(113.04)(5)=188.4 cm^3$[/tex]. |
Examine their arguments. Which statement explains whose argument is correct and why?
A. Paul's argument is correct; Manuel used the incorrect formula to find the volume of square pyramid X.
B. Paul's argument is correct; Manuel used the incorrect base area to find the volumes of square pyramid X and cone W.
C. Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid X.
D. Manuel's argument is correct; Paul used the incorrect base area to find the volumes of square pyramid X and cone W.
Answer (2)
Paul incorrectly calculates the base area of the cone, leading to incorrect volume calculations for both the cone and the pyramid.
Manuel correctly uses the formula π r 2 to calculate the base area of the cone.
Manuel
Explanation
Analyze Paul and Manuel's Arguments and Calculate Correct Values Let's analyze Paul and Manuel's arguments to determine who is correct and why.
Paul's Argument:
Base area of cone W: He uses a formula n ( d ) = n ( 12 ) = 37.68 c m 2 . This is incorrect. The correct formula is π r 2 .
Volume of cone W: He calculates 3 1 ( 37.68 ) ( 5 ) = 62.8 c m 3 . This is based on his incorrect base area.
Volume of square pyramid X: He calculates 3 1 ( 37.68 ) ( 5 ) = 62.8 c m 3 . This is also based on his incorrect base area.
Manuel's Argument:
Base area of cone W: He uses the formula π r 2 = π ( 6 2 ) = 113.04 c m 2 . This is the correct formula.
Volume of cone W: He calculates 3 1 ( 113.04 ) ( 5 ) = 188.4 c m 3 . This is based on his correct base area.
Volume of square pyramid X: He calculates 3 1 ( 113.04 ) ( 5 ) = 188.4 c m 3 . This is also based on his correct base area.
Now, let's calculate the correct base area and volume of the cone W.
The base area of cone W is given by the formula: A = π r 2 where r is the radius of the cone. Given that the radius is 6 cm: A = π ( 6 2 ) = 36 π ≈ 113.10 c m 2
The volume of cone W is given by the formula: V = 3 1 A h where A is the base area and h is the height. Given that the height is 5 cm: V = 3 1 ( 36 π ) ( 5 ) = 12 π ∗ 5 = 60 π ≈ 188.50 c m 3
Since the square pyramid X has the same base area and height as cone W, its volume is also: V = 3 1 A h = 3 1 ( 36 π ) ( 5 ) = 60 π ≈ 188.50 c m 3
Comparing the calculations, we can see that Manuel's calculations are closer to the correct values. Paul used an incorrect formula for the base area of the cone, which led to incorrect volumes for both the cone and the pyramid. Manuel's base area is more accurate (although rounded), and his volumes are also correct based on that base area.
Determine Whose Argument is Correct Manuel's argument is correct because he used the correct formula to find the base area of the cone, which is π r 2 . Paul used an incorrect formula, n ( d ) , which led to an incorrect base area and, consequently, incorrect volumes for both the cone and the square pyramid.
State the Correct Answer The correct answer is: Manuel's argument is correct; Paul used the incorrect base area to find the volumes of square pyramid X and cone W.
Examples
Understanding volumes of cones and pyramids is useful in architecture and engineering. For example, when designing roofs or structures with conical or pyramidal shapes, accurate volume calculations are essential for material estimation and structural stability. Knowing the correct formulas ensures efficient use of resources and safe construction.
Manuel's argument is correct because he uses the accurate formula for finding the base area of cone W, while Paul uses an incorrect approach. Consequently, Paul calculates incorrect volumes for both the cone and the square pyramid. The correct answer is option A: "Paul's argument is correct; Manuel used the incorrect base area to find the volumes of square pyramid X and cone W."
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