A cylindrical brass sleeve is to be shrink-tted over a brass shaft whose diameter is 3.212 cm at 0C. The diameter of the sleeve is 3.196 cm at 0C. (a) To what temperature must the sleeve be heated before it will slip over the shaft? (b) Alternatively, to what temperature must the shaft be cooled before it will slip into the sleeve?

(Midterm 2) Theory Lemmas, theorems and definitions Polynomials over a ring R Definition: Polynomials over a ring R (with coefficients in R) are expressions of type r0 1+r2+........n where, x is referred to an indeterminate (n≥0 an1 2…..,nare coefficients) subject to certain conventions. Lemma: All polynomials together with ‘+’, ‘.’ forms a ring called polynomial ring over R; denoted as R[x] (contains R) Lemma: Suppose R is an integral domain . Let f(x), g(x) ∈ R[x]. Then deg(f(x).g(x))=deg f(x) + deg g(x) Theorem: Suppose R is an integral domain, then the ring R[x] is an integral domain. Theorem: Let R be an integral domain. Let f(x)∈R[x] Then f(x) is a unit in R[x] Division algorithm for polynomials Let F be a field Let a(x), b(x) ∈ F[x], then there exists polynomials q(x), r(x) satisfying 1. a(x)=b(x)q(x)+r(x) 2. r(x)=0 or deg r(x)