Question

Problem 33b

Answer 1

Given information

Let be the value of the equipment at time .

Step-by-step explanation

Step 1.

 

We know from the given information that the rate of depreciation of the equipment is .

 

Because the derivative gives us the rate of change of a function, we can say that .

By taking the integral of both sides and applying the Fundamental Theorem of Calculus, we find that .

Because we defined as the value of the equipment, this equation tells us that the integral gives us the total loss of value at time  

We know that  represents the total depreciation in the value of the machine since the last overhaul. It is given that  is the cost incurred each time the machine is overhauled.

When we add the two, we get the total cost related to the maintenance of the machine. 

Therefore, the given function is  

If you don't overhaul the machine, it will cost the company in terms of the depreciation of the machine, but if you overhaul, there is a fixed cost related to it. This is why the term (the total cost related to the maintenance of the machine) makes sense, perform the maintenance, or won't, there is a cost associated with it.

Minimizing means that the company spends less overall and lower the expenditure, higher the profits.