Inventory Control
Jesaki Publishing sells 50,000 copies of a certain book each year. It costs the company $1 to store a book for one year. Each time that they print additional copies, it costs the company $1,000 to set up the presses. NOTE: We assume that the demand is uniform.
Let
x=number of books printed during each printing run number of printing runs
y=number of printing runs
The total setup cost for the year is ______ y.
Since we assume the demand is uniform, the number of books in storage between printing runs will decrease from x to ____.
Since it costs $1 to store a book for one year, the total storage cost is _____ x.
Note: The average number in storage for each day is x/2.
Enter your answer as a decimal.
The total cost is the sum of the setup cost and storage cost, so
C=____y+_______x.
In order to write the total cost as a function of one variable, we must find a relationship between x and y. Since Jesaki prints x books in each of the y printing runs, the total number of books printed is xy, so we must have
xy= .
We can use this to express y as a function of x.
Since x is the number of books printed in each printing run, x must satisfy 1≤x≤ ____. .
In other words, x is in the closed interval [a,b], where a=1 and b=
.
Using the calculations above, we can express the total cost C(x) as a function of x, with the restriction on x given in the previous problem.
Find the critical number of C(x) by solving C′(x)=0.
NOTE: Because of the restriction on x, there is exactly one critical number c.
There is only one critical number c in the interval, and the cost function C(x) is continuous.
Since C′(c) [ Select ] ["< 0", "> 0", "≤ 0", "≠ 0", "≥ 0", "= 0"]
and C″(c) [ Select ] ["> 0", "≠ 0", "< 0", "= 0", "≤ 0", "≥ 0"] ,
we can use the [ Select ] ["First-Derivative Test for Absolute Extrema on an Interval", "Second-Derivative Test for Absolute Extrema on an Interval", "Extreme Value Theorem", "Chain Rule", "L'Hopital's Rule"]
to conclude that C(c) is the [ Select ] ["absolute minimum", "intercept", "asymptote", "absolute maximum"] of the cost function on the interval I.
How many books should be produced during each printing run to minimize total cost?
___________ books
How many printing runs should be done?
____________printing runs
What is the minimum total cost? $________