Question

question 10 multiple choice worth 1 point) (06.01 hc) 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? paul 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(8²) = 200.96 cm². the volume of cone w is (area of base)(h) = 1/3 (200.96)(5) = 334.93 cm³. the volume of square pyramid x is 1/3 (area of base)(h) = 1/3 (200.96)(5) = 334.93 cm³. manuel 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(8²) = 200.96 cm². the volume of cone w is 1/3 (area of base)(h) = 1/3 (200.96)(5) = 334.93 cm³. the volume of square pyramid x is (area of base)(h) = (200.96)(5) = 1,004.8 cm³. o pauls argument is correct, manuel used the incorrect formula to find the volume of square pyramid x. o pauls argument is correct, manuel used the incorrect base area to find the volume of square pyramid x. o manuels argument is correct, paul used the incorrect formula to find the volume of square pyramid x. o manuels 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 \( V_{cone}=\frac{1}{3}Bh\) (where \( B\) is the base - area and \( h\) is the height), and the volume formula for a square pyramid is \( V_{pyramid}=\frac{1}{3}Bh\).

Step2: Analyze Paul's work

Paul used the formula \( V_{cone}=\frac{1}{3}Bh\) (where \( B = \pi r^{2}=\pi(8)^{2}=200.96\space cm^{2}\) and \( h = 5\space cm\)), so \( V_{cone}=\frac{1}{3}(200.96)(5)=334.93\space cm^{3}\). Since \( V_{pyramid}=\frac{1}{3}Bh\) (with the same \( B\) and \( h\) as the cone), Paul's formula application for the cone and the fact that the pyramid has the same \( B\) and \( h\) is correct.

Step3: Analyze Manuel's work

Manuel made a mistake. The formula for the volume of a square pyramid is \( V=\frac{1}{3}Bh\). Manuel calculated \( V=(200.96)(5)=1004.8\space cm^{3}\), which is the formula for the volume of a prism (\( V = Bh\)) instead of a pyramid.

Answer:

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