Chapter 2.4, Problem 59E

Replacement time. An office copier has an initial price of $\text{\$2,500}$. A service contract costs $\text{\$200}$ for the first year and increases $\text{\$50}$ per year thereafter. It can be shown that the total cost of the copier after $n$ years is given by

$C\left(n\right)=2,500+175n+25n^2$

The average cost per year for $n$ years is given by $\bar{C}\left(n\right)=C\left(n\right)/n$.

(A) Find the rational function $\bar{C}$.

(B) Sketch a graph of $\bar{C}\text{ for }2\le n\le 20$.

(C) When the average cost per year at a minimum, and what is is the minimum average annual cost? [Hint: Refer to the sketch in part (B) and evaluate $C\left(n\right)$ at appropriate integer values until a minimum value is found.] The time when the average cost is minimum is frequently referred to as the replacement time for the piece of equipment.

(D) Graph the average cost function $C$ on a graphing calculator and use an appropriate command to find when the average annual cost is at a minimum.