Question

A car company wants to have a new factory to expand their production. The company is considering between two options: 1. They can build a new factory with a total cost of 50 million dollars. 2. They can spend 5 million dollars on buying new equipment and rent an existing factory with a cost of 1.5 million dollars per month. At the end of every month, the company has total sales of 2 million dollars with a cost of $400,000 incurred during that month. (a) Find two linear cost functions corresponding to two scenarios and the revenue function. (b) If the company wants to start getting profit as soon as possible, which option the company should choose? (c) Graph the two cost functions and revenue function and interpret the result you found in part (b).

Answer 1

a)
Let n represents the number of months, company runs.
Net Revenue = TR=(Sales-Cost)*No. of months=(2000000-40000)*n=1960000n
Option 1= Built a new factory
Total Cost=TC1=$50,000,000

Option 2= Buy some equipment at $5 million and Rent an existing factory
Total Cost=TC2=Equipment cost + Monthly cost
Total Cost=TC2=5000000+1500000*n
b)
Let us calculate the number of months by which company will be able to recover all costs
Option 1= Built a new factory
Set Net Revenue=Total Cost
1960000n=50000000
n=50000000/1960000=25.51 months

Option 2= Buy some equipment at $5 million and Rent an existing factory
Set Net Revenue=Total Cost
1960000n=5000000+1500000*n
n=5000000/(1960000-1500000)= 10.87 months

It is clear that Option 2 will start giving profits early.

c)
With the help of given total cost and total revenue functions, we can make the following schedule

n	TC1	TC2	TR
0	50000000	5000000	0
5	50000000	12500000	9800000
10	50000000	20000000	19600000
15	50000000	27500000	29400000
20	50000000	35000000	39200000
25	50000000	42500000	49000000
30	50000000	50000000	58800000

TR =1960000n
TC1=$50,000,000
TC2=5000000+1500000*n

70000000 TC2 20000000 S1015 202S 30 35 No. of months, n

Looking at the graph, we can make that
1)   Option 2 will give higher profit when n>10.87 months but n<25.51 months
2)   Option 1 will give higher profits when n>25.51 months