Calculus question

Question

A bicycle plant assembles 1000 bicycles per month. Each production run costs $1200, and it costs $20 to store a bicycle for a month. How many production runs should the plant use to minimize inventory costs? How many bicycles are assembled in each production run? 

Answer 1

The inventory cost can be seen as:

C=1200x+20(1000/x)

Differentiate

dC/dT=1200-20000/(x²)

Set dC/dT equal to 0 in order to find maxima/minima, and solve for x

0=1200-20000/(x²)

-1200=-20000/(x²)

-1200x²=-20000

x²⁼16.667

x=4.08 or -4.08

Since you cannot produce a negative number of bikes nor a partial bike, we will use the positive one and round

x=4 production runs

bikes per month divided by number of production runs=number of bikes per run

1000/4=250 bikes per run

You can check this by plugging in 4 to the equation, and then plugging in a number greater than 4 and one less than 4

1200*4+20000/4=$9800

1200*5+20000/5=$10000

1200*3+20000/3=$10266.7

The inventory cost is greater for both values, so 4 does produce the minimum inventory cost.