Question

Each machine at a certain factory can produce 90 units per hour. The setup cost is 20 dollars for each machine and the operating cost is 26 dollars per hour (total, not 26 dollars per machine per hour). You would like to know how many machines should be used to produce 40000 units, with the goal of minimizing production costs.

First, find a formula for the total cost in terms of the number of machines, n:_______
TC = ______
machines for a total cost of The minimum total cost is achieved when using dollars.

Answer 1

a)
`Total Cost=20n+(((40000)/(90)*26))/(n)`

b)
`n=24`

Explanation:

From the question we are told that:

Rate r=90 units per hour

Setup cost =20

Operating Cost =26

Units=40000

Generally the equation for Total cost is mathematically given by

`Total Cost=20n+(((40000)/(90)*26))/(n)`

`T_n=20n+(11556)/(n)\\\\T_n=(20n^2+11556)/(n).....equ 1`

Differentiating

`T_n'=(n(40n)-(40n^2+11556))/(n_2)\\\\T_n'=(20n^2-11556)/(n^2).....equ 2`

Equating equ 1 to zero

`0=(20n^2+11556)/(n)`

`n=24`

Therefore

Substituting n

For Equ 1

`T_n=(20(24)^2+11556)/(24)`

F(n)>0

For Equ 2

`T_n'=(20(24)^2-11556)/(24^2)`

F(n)'<0