Question

Foggy Optics, Inc., makes laboratory microscopes. Setting up each production run costs $2500. Insurance costs, based on the average number of micro- scopes in the warehouse, amount to $20 per microscope per year. Storage costs, based on the maximum number of microscopes in the warehouse, amount to $15 per microscope per year. If the company expects to sell 1600 microscopes at a fairly uniform rate throughout the year, determine the number of production runs that will mini- mize the company’s overall expenses.

Answer 1

Answer: 4

Step-by-step explanation:

Let N be the total number of runs.

The microscopes per year will be equal to the total number of runs * microscopes per run

total number of microscopes = total number of runs * microscopes per run

now we have each run to cost 2500, therefore:

run costs = 2500 * total runs = 2500*N

we also have insurance and storage costs (note the average thats why we divide by 2):

storage costs = 20* microscopes per run /2 = 20*N/2

insurance costs = 15 * microscopes per run = 15*N

the costs are:

costs = run costs + storage + insurance

now the comany sells 1600 microscopes a year, so in order to figure out the number of microscopes per run we divide 1600 by the total number of runs N and replace in the costs

costs = 2500N + 40000/N => C(N) = 2500N + 40000/N

now derivate the function with respect to N and equal to 0:

$\frac{d}{dN}(2500N + 40000/N) = 2500(1-16/N^2) \\\\ 2500(1-16/N^2) = 0$

this gives you two solutions, N = -4 which is not eligible (you cant make negative microscopes lol) and N = 4 which is the answer

have a nice day