Question

question 5 (multiple choice worth 1 points)cone w has a radius of 8 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?paulthe 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 $pi(r^2)=pi(8^2)=200.96 \text{cm}^2$. the volume of cone w is $\frac{1}{3}(\text{area of base})(h)=\frac{1}{3}(200.96)(5)=334.93 \text{cm}^3$. the volume of square pyramid x is $\frac{1}{3}(\text{area of base})(h)=\frac{1}{3}(200.96)(5)=334.93 \text{cm}^3$manuelthe 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 $pi(r^2)=pi(8^2)=200.96 \text{cm}^2$. the volume of cone w is $\frac{1}{3}(\text{area of base})(h)=\frac{1}{3}(200.96)(5)=334.93 \text{cm}^3$. the volume of square pyramid x is $(\text{area of base})(h)=(200.96)(5)=1,004.8 \text{cm}^3$pauls argument is correct. manuel 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.manuels argument is correct. paul used the incorrect formula to find the volume of square pyramid x.manuels argument is correct. paul used the incorrect base area to find the volume of square pyramid x

Explanation:

Step1: Recall cone volume formula

Volume of cone: $V_{cone} = \frac{1}{3} \times \text{base area} \times h$

Step2: Recall pyramid volume formula

Volume of pyramid: $V_{pyramid} = \frac{1}{3} \times \text{base area} \times h$

Step3: Analyze Paul's calculation

Cone base area: $\pi r^2 = \pi(8^2) = 200.96\ \text{cm}^2$
$V_{cone} = \frac{1}{3} \times 200.96 \times 5 = 334.93\ \text{cm}^3$
Pyramid has same base area/height: $V_{pyramid} = \frac{1}{3} \times 200.96 \times 5 = 334.93\ \text{cm}^3$

Step4: Analyze Manuel's error

Manuel omitted the $\frac{1}{3}$ term for the pyramid: $V_{pyramid} = 200.96 \times 5 = 1004.8\ \text{cm}^3$, which uses the wrong formula.

Answer:

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