Chapter 5.4, Problem 32E

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

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

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

(b) Suppose that

$f(t)=\{\begin{array}{lll}\frac{V}{15}-\frac{V}{450}t\hfill & \text{if}\hfill & 0<t\le 30\hfill \\ 0\hfill & \text{if}\hfill & t>30\hfill \end{array}$

and

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

Determine the length of time T for the total depreciation $D(t)={\displaystyle {\int }_0^{t}f(s)ds}$ 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.