the diameter of a sphere in centimeters Let a = 360 and b = 702. Find the prime
ID: 1527063 • Letter: T
Question
the diameter of a sphere in centimeters Let a = 360 and b = 702. Find the prime factorizations of a and b Use Corollary 1.3.5 to find the prime factorizations of god(a, b) and lcm(a, b) Prove there are no positive integers m and n such that m^3 = 2n^3. Remark: A classical problem of antique mathematics was to construct a cube with twice the volume of some original cube. The above statement essentially shows that the ratio of the side lengths of the two cubes, namely 3 Squareroot 2, is an irrational number. Below are two false statements and their erroneous proofs (by induction). For each of them, find a counterexample to the statement, determine for which value(s) of n the inductive proof breaks down, and describe the logical mistake(s) in that step (e.g., by pointing out the incorrect sentence(s). Statement: All elements in an arbitrary finite set of integers have the same absolute value (in symbols: if X is a finite set of integers, then for any integers m and m' in X, we have |m| = |m'|). Proof: We use induction on the size of the finite set: more precisely, define the statement P(n) to be "all elements in an arbitrary set of n integers have the same absolute value", and we use induction on n to show P(n) holds for any n epsilon P. Base case: When n = 1, the statement P(n) holds trivially because our set has only one element Induction step: Suppose P(k) holds, i.e., all elements in an arbitrary set of k integers have the same absolute value. Then to prove P(k+ 1), we need to show if X is a set of k + 1 integers, then for any integers m and m' in X, we have |m| = |m'|. Now given such a set X of k + 1 integers and given such integers m and m' in X, we discuss two possibilities: If m = m', then clearly |m| = |m'|. On the other hand, if m = m', then we define X_1 = {x epsilon X: x = m} and X_2 = {x epsilon X: x = x'}. In other words, X_1 is the set of all integers in X except m, and X_2 is the set of all integers in X except m'. Then we pick a third integer 1 in X which is different from m and m'. By the definition of X_1 and X_2, we know that 1 and m' belong to X_1, and 1 and m belong to X_2. Since both X_1 and X_2 contain exactly k integers, thus by our inductive assumption, all elements of X_1 have the same absolute value and so do all elements of X_2. Hence we have |m| = |l| = |m'|. In any case, any integers m and m' in X always satisfy |m| = |m|. This shows that P(k + 1) also holds. Therefore by the induction principle, all elements in an arbitrary finite set of integers have the same absolute value.Explanation / Answer
a) 1 inch = 2.54 cm
radius = diameter / 2
= 4.71 / 2 = 2.355 inch
= 2.355*2.54 = 5.9817 cm
b) surface area = 4pi*r^2
= 4pi*5.9817^2 = 449.63 cm2
v) volume = 4/3 pi*r^3
= 4/3*pi*5.9817^3
= 896.52 cm3
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