Homework Clinic
Mathematics Clinic => Calculus => Topic started by: cnetterville on Dec 18, 2021
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A manufacturer uses raw materials to produce p products each day. Suppose that each delivery of a particular material is $d, whereas the storage of that material is x dollars per unit stored per day. (One unit is the amount required to produce one product). How much should be delivered every x days to minimize the average daily cost in the production cycle between deliveries?
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If he asks for a delivery every x days, then he must order (px) to have enough material for that delivery cycle. The average amount in storage is approximately one-half of the delivery amount, or
. Thus, the cost of delivery and storage for each cycle is approximately
Cost per cycle = delivery costs + storage costs
Cost per cycle = d + ∙ x
We compute the average daily cost c(x) by dividing the cost per cycle by the number of days x in the cycle.
c(x) = 
+
We find the critical points by determining where the derivative is equal to zero.
c'(x) = - 
+ 
= 0
x = ± 
Therefore, an absolute minimum occurs at days.