A company models the average production cost per item by the function C

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

A company models the average production cost per item by the function

C(a)=50+\frac{10a}{a-2}, \quad a>0,\; a\neq 2,

where

a

is the number of items (in thousands) produced in one day and

C(a)

is measured in U.S. dollars. (a) Identify all real-number inputs that are not in the domain of

C

and explain their exclusion in terms of production. (b) Compute

\displaystyle \lim_{a\to 2^-}C(a)

and

\displaystyle \lim_{a\to 2^+}C(a)

. Interpret each limit in the business context. (c) Based on part (b), discuss whether

a=2

corresponds to a vertical asymptote of the graph of

C

. Provide mathematical justification. (d) The company plans never to exceed 10 (thousand) items per day. Determine the interval of

a

-values on which the model predicts decreasing cost. Be sure to exclude any intervals made meaningless by the asymptotic behavior. Explain your reasoning fully.

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