1. ## Optimization

McDuff Preserves expects to bottle and sell 7000000 32-oz jars of jam. The company orders its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles is $700, and the cost of storing each empty bottle for a year is$.80. How many orders should McDuff place per year and how many bottles should be in each shipment if the ordering and storage costs are to be minimized? (Assume that each shipment of bottles is used up before the next shipment arrives.)

2. Hello,Tim!

McDuff Preserves expects to bottle and sell 7,000,000 32-oz jars of jam. The company orders
its containers from Consolidated Bottle Company. The cost of ordering a shipment of bottles
is $700 per shipment, and the cost of storing each empty bottle for a year is$0.80.
How many orders should McDuff place per year and how many bottles should be in each
shipment if the ordering and storage costs are to be minimized? (Assume that each
shipment of bottles is used up before the next shipment arrives.)

Let $\displaystyle N$ = the number of shipments.
The cost of shipping is: .$\displaystyle 700N$ dollars.

Then there are: .$\displaystyle \frac{7,000,000}{N}$ bottles per shipment.
The cost of storage is: .$\displaystyle (0.08)\left(\frac{7,000,000}{N}\right) \:=\:\frac{560,000}{N}$ dollars.

Hence, the cost is: .$\displaystyle C \;=\;700N + 560,000N^{-1}$

Differentiate and equate to zero: .$\displaystyle C\,' \;=\;700 - 560,000N^{-2} \;=\;0$

Multiply by $\displaystyle N^2\!:\;\;\;700N^2 - 560,000 \:=\:0\quad\Rightarrow\quad N^2 \:=\:800$

Hence: .$\displaystyle N \;=\;\sqrt{800} \;=\;20\sqrt{2}\;\approx\;28.2842...$

Therefore, McDuff should order 28 shipments
. . . . . . . with 250,000 bottles per shipment.