Astudent memorized the statement the product of any rational number and any irrational number is irrational. label this statement as true or false, and explain your reasoning.

a) true. for example, 3 ·5 is irrational, 7 · 2 3 is irrational, and (π)·(−4) is irrational.

b) false. suppose an irrational times a rational = rational, then irrational = rational rational . this means irrational = rational, which is a contradiction. thus an irrational times a rational = rational.

c) true. suppose an irrational times a rational = rational, then irrational = rational rational . this means irrational = rational, which is a contradiction. thus an irrational times a rational = irrational.

d) false. the rational number cannot be zero. suppose an irrational times a nonzero rational = rational, then irrational = rational nonzero rational . this means irrational = rational, which is a contradiction. thus an irrational times a nonzero rational = irrational.

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 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(82) = 200.96 cm2. the volume of cone w is one third(area of base)(h) = one third third(200.96)(5) = 334.93 cm3. the volume of square pyramid x is one third(area of base)(h) = one third(200.96)(5) = 334.93 cm3. 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 one third(area of base)(h) = one third(200.96)(5) = 334.93 cm3. the volume of square pyramid x is (area of base)(h) = (200.96)(5) = 1,004.8 cm3. paul's argument is correct; manuel used the incorrect formula to find the volume of square pyramid x. paul's argument is correct; manuel used the incorrect base area to find the volume of square pyramid x. manuel's argument is correct; paul used the incorrect formula to find the volume of square pyramid x. manuel's argument is correct; paul used the incorrect base area to find the volume of square pyramid x.