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.

Let $$C(t)=\frac{1}{t}\int_{0}^{\,t} {[f(s)+g(s)]}\,ds$$ It can be shown that the critical numbers of $C$ occur at the numbers $t$ where $C(t)=f(t)+g(t).$

(a) Suppose that $$f(t)=\begin{cases} \displaystyle\frac{V}{15}-\frac{V}{450}t & \text{ if } 0 < t \le 30 \\ 0 & \text{ if } t > 30 \end{cases}$$ and $$g(t)=\frac{Vt^2}{12900}\qquad t>0$$ Determine the length of time $T$ (in months) for the total depreciation $D(t)=\int_{0}^{\,t} {f(s)}\,ds$ to equal the initial value $V$.

$T$ = months.

(b) Determine the absolute minimum of $C$ on $(0, T]$. Your answer may depend on $V$.

Absolute minimum of $C$ is