a bookstore has an annual demand for 117000 copies of a cert

a bookstore has an annual demand for 117,000 copies of a certain book. it cost $70 to place an order and .20cents to store one copy for one year. Find optimum number of copies per order. explain answer and what key words you used to derive the solution.

Solution

I will make a couple assumptions: first that the nth book is bought precisely when n/117000 of the year has passed, and second that to store a book a fraction x of a year it costs x * 0.20 dollars. Hopefully these are reasonable. I do not know how we would solve it otherwise.

So, for example, if we ordered all of the books on the precise beginning of the year it would cost us $70 + $0.20 * (1/117000 + ... + 117000/117000) = $70 + $0.20 * 117001/2 = $11770.1. I used fact that the formula for the sum of the first n integers is n(n+1)/2.

Now let\'s come up with a general formula for the cost as a function of the number of orders. If we have 117000/k orders of k books each (for now assume k evenly divides 117000) then the cost is $70 * (117000/k) + $0.20 * (117000/k) * [1/117000 + ... + k/117000]. This is because there are 117000/n books ordered in each order. Simifying this (again using the formula for the sum of the first n integers) we get:

$70 * (117000/k) + $0.20 * (117000/k) * k(k+1)/117000 = $7020000/k + $0.20 * (k+1)
Taking the derivative with respect to k and setting it equal to 0 we get:
-$7020000/k^2 + $0.20 = 0
Simplifying we get k^2 = 35100000, so k = 5924.5, around 16 orders a day...
If we want the number of orders to be a divisor of 117000, then we should check the two divisors of 117000 that are closest to 5924.5.

Hope that helps.