How does someone like Bill Gates becomes so wealthy? We all know that poor sales mean financial ruin, while stratospheric sales can create similarly spectacular profits. In this activity, you will create a quantitative model illustrating this, and use it to analyze and forecast

An employer pays $3,000 per month for rent office space, and pays a total of 12 employees (including yourself) $1,000 per month plus a bonus of $10 for each unit of work completed by the worker that month. Suppose also that your employer makes $18 (gross) per unit of work by an employee.

1. Find your monthly salary S(x) as a function of x, the number of units of work you complete that month. S(x)=1000+10x

2. If all of your co-workers produce/sell (on average) x units per month, find the monthly profit P(x) of your employer Phil Bates as a function of x.
   P(x)=18(12x) - 3000 - 12(1000+10x) = 216x - 15000 - 120x = 96x-15000

3. Suppose you complete 200 units per month. Find your monthly paycheck. S(200) = 3,000

4. Suppose all of your co-workers (on average) also complete 200 units per month. Find your boss Phil Bates' profits per month. P(200) = (96)(200) - 15,000 = $4,200

5. Suppose you complete 300 units per month. Find your monthly paycheck. And also, supposing all of your co-workers (on average) also complete 300 units per month, find Phil Bates' profits on such months. S(300) = $4,000 ; P(300) = $13,800

6. How bad a month (how few a number of units of work completed) does it have to be for your employer to make the same monthly salary you do? Set S(x) = P(x) to get 1000+10x = 96x-15,000 thus 16,000 = 86x, so x = 186 plus change; round down: x=186

7. Due to the his revenues from the units of work you completed (service calls, sales, computer programs written, etc), your employer Phil Bates (following his wealthier cousin Bill) is able to now rent a bigger office ($10,000 per month), and employ 50 employees, each one still getting $1,000 per month plus $10 per unit of work completed, and again each unit of work completed makes $18 (gross) for Phil Bates.
   A new P(x) is 18(50x) - 10,000 - 50(1000+10x) = 400x - 60,000

   How many units (x) would you and your co-workers have to complete each month before Phil's (monthly) profits are double your monthly wage? Ten times your monthly wage?
   Set P(x) = 2S(x) to get 400x-60,000 = 2000+20x or x=(62,000)/380 so x=163 plus change; round up if you Phil is to have at least twice the salary, so 164. For second part, replace "2" by "10" and a similar calculatoin gives x=233 plus change so round up to 234.

   What if you produce/sell x=200 or 300 as in the earlier scenarios?
   If x=300 then (still using the new P(x)), P(300) = $60,000 per month for Phil Bates ($720,000/year in profits); At x=2,650 units per month, Phil would have profits of $1,000,000 per month.