Question

cone w has a radius of 10 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 how the volumes of cone w and square pyramid x are related. examine their arguments. which statement explains whose argument is correct, and why? paul: the volume of square pyramid x is three times the volume of cone w. this can be proven by finding the base area and volume of cone w, along with the volume of square pyramid x. the base area of cone w is π(r²)=π(10²)=314 cm². the volume of cone w is 1/3 (area of base)(h)=1/3 (314)(5)=523.33 cm³. the volume of square pyramid x is (area of base)(h)=(314)(5)=1,570 cm³. 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 w, along with the volume of square pyramid x. the base area of cone w is π(r²)=π(10²)=314 cm². the volume of cone w is 1/3 (area of base)(h)=1/3 (314)(5)=523.33 cm³. the volume of square pyramid x is 1/3 (area of base)(h)=1/3 (314)(5)=523.33 cm³. pauls argument is correct; manuel used the incorrect formula to find the volume of square pyramid x. manuels argument is correct; paul used the incorrect formula to find the volume of square pyramid x. pauls argument is correct; manuel used the incorrect base area to find the volume of square pyramid x. pauls argument is correct; paul used the incorrect base area to find the volume of square pyramid x.

Explanation:

Step1: Recall volume formulas

The volume formula for a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h$, and for a square - pyramid is $V_{pyramid}=\frac{1}{3}Bh$, where $B$ is the base - area and $h$ is the height. Given that the base - area of the cone $B_{cone}=\pi r^{2}$ (with $r = 10$ cm) and $h = 5$ cm, and the base - area of the square pyramid is the same as the base - area of the cone, and height is also the same.

Step2: Calculate base - area of the cone

The base - area of the cone $B=\pi r^{2}$, substituting $r = 10$ cm, we get $B=\pi\times(10)^{2}=100\pi\approx314$ $cm^{2}$.

Step3: Calculate volume of the cone

Using the volume formula for the cone $V_{cone}=\frac{1}{3}Bh=\frac{1}{3}\times314\times5=\frac{1570}{3}\approx523.33$ $cm^{3}$.

Step4: Calculate volume of the square pyramid

Using the volume formula for the square pyramid $V_{pyramid}=\frac{1}{3}Bh$, with $B = 314$ $cm^{2}$ and $h = 5$ cm, we get $V_{pyramid}=\frac{1}{3}\times314\times5=\frac{1570}{3}\approx523.33$ $cm^{3}$.

Step5: Analyze Paul's and Manuel's arguments

Paul says the volume of the square pyramid $X$ is three times the volume of cone $W$, which is incorrect. Manuel says the volume of square pyramid $X$ is equal to the volume of cone $W$, which is correct because when the base - area and height are the same, $V_{cone}=\frac{1}{3}Bh$ and $V_{pyramid}=\frac{1}{3}Bh$.

Answer:

Manuel's argument is correct; Paul used the incorrect formula to find the volume of square pyramid $X$.