Chapter 4.3, Problem 78E

A high-tech company purchases a new computing system whose initial value is V. The system will depreciate at the rate f = f(t) and will accumulate maintenance costs at the rate g = g(t), where t is the time measured in months. The company wants to determine the optimal time to replace the system.

(a) Let

$C\left(t\right)=\frac{1}{t}{\displaystyle {\int }_0^t\left[f\left(s\right)+g\left(s\right)\right]ds}$

Show that the critical numbers of C occur at the numbers t where C(t) = f(t) + g(t).

(b) Suppose that

$\begin{array}{l}f\left(t\right)\{\begin{array}{l}\frac{V}{15}-\frac{V}{450}t\text{ if 0 < }t⩽30\\ 0\text{ if }t>30\end{array}\\ \\ \end{array}$

and

$g\left(t\right)=\frac{V{t}^2}{12,900}t>0$

Determine the length of time T for the total depreciation $D\left(t\right)={\displaystyle {\int }_0^{t}f\left(s\right)}ds$to equal the initial value V.

(c) Determine the absolute minimum of C on (0, T].

(d) Sketch the graphs of C and f + g in the same coordinate system, and verify the result in part (a) in this case.