?A manufacturing company owns a major piece of equipment that depreciates at the

Chapter 4, Problem 95

(choose chapter or problem)

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate \(f(t)\), where \(t\) is the time measured in months since its last overhaul. Because a fixed \(\operatorname{cost} A\) is incurred each time the machine is overhauled, the company wants to determine the optimal time \(T\) (in months) between overhauls.

(a) Explain why \(\int_{0}^{t} f(s) d s\) represents the loss in value of the machine over the period of time \(t\) since the last overhaul.

(b) Let \(C=C(t)\) be given by

\(C(t)=\frac{1}{t}\left[A+\int_{0}^{t} f(s) d s\right]\)

What does \(C\) represent and why would the company want to minimize \(C\) ?

(c) Show that \(C\) has a minimum value at the numbers \(t=T\). where \(C(T)=f(T)\).

Equation Transcription:

Text Transcription:

f(t)

t

A

T

int_{0}^{t} f(s) ds

t

C=C(t)

C(t) = frac{1}{t} [A+\int_{0}^{t} f(s) ds]

C

C

C

t =T

C(T)  =f  (T)