A computer software company sells 20,000 copies of a certain computer game each

Question

A computer software company sells 20,000 copies of a certain computer game each year. It costs the company $1.00 to store each copy of the game for one year. Each time it must produce additional copies, it costs the company $625 to set up production. How many copies of the game should the company produce during each production run in order to minimize its total storage and set-up costs? 5000 copies in 4 production runs 10,000 copies in 2 production runs 20,000 copies in 1 production run 4000 copies in 5 production runs

Explanation / Answer

The total cost is
C(x) = 625*20000 (1/x) + (1/2)x

C (x) = 12500000x^1 + 0.5x.
We compute
C'(x) = 12500000x^2 + 0.5.
Thus we have a critical point at x = 5000. Since
C"(x)= 25000000x^3

we have C(5000) > 0.

Then by the Second Derivative Test C(x) has a minimum, when x = 5000. Since the company wishes to sell 20000 copies, this means we need
20000/5000 = 4 production runs.

Answere a)

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