Please help, as it is prep for final exam... ONLY use written instructions during show work..

Assume you have one copy machine. The machine, being used 24 hour everyday, will wear out eventually, and you will have to buy anew machine periodically.This problem is aboutfinding an optimal timebetween twooverhauls(tossing away old one and getting new one). Here'optimal' means incurring aminimum monthly cost.There are two kinds of cost incurred during a period from an overhaul to the next.

• Fixed cost: This is the price of the copy machine to replace the old one.
• Monthly depreciation:This is the depreciation of the value of the machine that incur in a month. This will vary month to month and in general increases as the machine gets older.

(a) Suppose the machine's depreciation rate isf(s)=100sqrt s dollars per monthat the time when it iss months old. Find the total depreciation that incur during the firsttmonths after an overhaul. Your answer should be a function oft of course.

(b) Now, suppose the price of a brand new machine isA =$3,000. This is the fixed cost.The total depreciation you answeredin (a) can be also considered asa cost. Add these two costs to form the total cost that incur during the time interval[0,t] Graph this function.Label the horizontal axis as "month" and the vertical axis as"total cost".Hand drawing is OK!But, make sure the concavity is correct.Label the intercept with a specific dollar amount

(c) Explain from a copy shop owner's perspective why the "Total Cost" is not what you need to pay attention. What is it that you would like to minimize?

(d) [ 5 points] LetC denote theAVERAGE MONTHLY COST.Write the average cost function,C=C(t),fort0 based onthe specific scenario stated in (a) and (b) above.This is the monthly average cost function.To be more specific. This is the average of monthly coststhat incur during theperiod given by the interval[0,t].

(e) Now we are all prepared to find the optimal "stopping" time.Find the minimum of the functionC(t). First, find the critical numberT(Exact value! Do not approximate.) on the interval0Write the conclusion:"In order to minimize the average monthly cost, the machine should be overhauled every ________ months."

(f) Reuse your graph in (b).Copy it hereand do the following. (i) pick upa few points on the graph and draw lines from the origin to these points. (ii) Interpret the slopes of these lines.

(g) In this problem,you will not be using the specific functionf(x) or the specific valueA

given in parts (a) and (b).It is a completely general question. Use the general formulafor the Monthly Cost function provided there.Show that ifTis a critical numberfor this function, thenC(T)=f(T).

(h) OPTIONAL: On your graph for the total cost function (in (b)), draw a line through the origin whose slope isf(T) and explain why the slope of that line should be f(T)