A manufacturing company owns a major piece of equip -ment that depreciates at the (continuous) rate f= f(t) ,where is the time measured in months since its last overhaul.Because a fixed cost is incurred each time themachine is overhauled, the company wants to determine theoptimal time T (in months) between overhauls.(a) Explain why ∫0t f(s) ds represents the loss in value ofthe machine over the period of time t since the lastoverhaul. (b) Let C =C(t) be given by C(t) =1/t[A + ∫0t f(s) ds]What does C represent and why would the companywant to minimize C ?(c) Show that has a minimum value at the numbers t =Twhere C(T) =f(T).
A manufacturing company owns a major piece of equip -ment that depreciates at the (continuous) rate f= f(t) ,where is the time measured in months since its last overhaul.Because a fixed cost is incurred each time themachine is overhauled, the company wants to determine theoptimal time T (in months) between overhauls.(a) Explain why ∫0t f(s) ds represents the loss in value ofthe machine over the period of time t since the lastoverhaul. (b) Let C =C(t) be given by C(t) =1/t[A + ∫0t f(s) ds]What does C represent and why would the companywant to minimize C ?(c) Show that has a minimum value at the numbers t =Twhere C(T) =f(T).
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.CR: Chapter 3 Review
Problem 8CR
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Question
A manufacturing company owns a major piece of equip -
ment that depreciates at the (continuous) rate f= f(t) ,
where is the time measured in months since its last overhaul.
Because a fixed cost is incurred each time the
machine is overhauled, the company wants to determine the
optimal time T (in months) between overhauls.
(a) Explain why ∫0t f(s) ds 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) =1/t[A + ∫0t f(s) ds]
What does C represent and why would the company
want to minimize C ?
(c) Show that has a minimum value at the numbers t =T
where C(T) =f(T).
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