Question

Calc (with business applications) Using derivatives to find the cheapest order size. Not adding up correctly.

Answer 1

I am using a piecewise function to find the quantity ordered that saves the most money. The formula would be \[(base cost * total needed) + (shipping cost * total needed/quantity) + (storage cost * average quantity)\]
the equation is \[525(15)+200(\frac{ 15 }{ q })+7.5(\frac{ q }{ 2 }) if 0<q<15 \] or
\[505(15)+200(\frac{ 15 }{ q })+7.5(\frac{ q }{ 2 }) if 15≤q\]

Answer 2

First, plotted the equation to see which (of the piecewise) would have the lowest cost, then I found the derivative, set it equal to zero and solved for q, which gave me :

q=63.24555320 for a cost of $38,349.34.

this is the answer that my teacher wants, but when I actually add these orders together,[(75000/63.24555320 = total number of orders which is 1185.854123) 1185.85... * $38,349.34] which gives me the total cost of all the orders which is
$45,443,967.90
and that leaves the last order, which would be $38,353.978 (remaining quantity needed times price of the first piece wise equation) I get a grand total of $45,482,321.88

The problem with all of this is, when I place an order of all 75,000 at once, it only costs me $319,125.20