Jack was named manager of the month at Sylvania Wire Company due to his hiring study. The study showed that each of the 30 salespersons he supervised averaged $50,000 in sales each month, and that for each additional salesperson he would hire, the average sales would decrease by $1,000 per salesperson.
a) Write a function for Jack's study showing the total sales as a function of the number of additional salesperson Jack hires.
b) Use your function to determine how many salespersons Jack should hire to maximize the income from sales. What is the maximum sales Jack's company can anticipate according to the model?
These are always a little difficult for me to think of the correct function!!!
a. Call x the number of additional people hired ........ so we have
Sales = (30 + x) (50000 - 1000x) = -1000x^2 + 20000x + 1500000
b. To find the number of additional hires that will maximize revenue......we can use
x = -20000 / [ -2 * 1000] = -20000 / -2000 = 10 additional people hired will maximize the revenue
The max sales will be (30 + 10) (50000 - 1000 (10) ) = 40 * 40000 = $ 1,600,000
Sorry.....I should have explained that....!!!!
In a quadratic that has the form ax^2 + bx + c, the maximum [ or minimum] is produced at the x value given by :
-b / [ 2a]
So ......using our function -1000x^2 + 20000x + 1500000
a = -1000 and b = 20000
So....the x value [ additional employees hired ] that produces the max revenue is
-20000 / [ 2 ( -10000) ] = -20000 / -2000 = 10
Does that help ???