Can someone help with this calculus problem?

Question

-A man has 61 clients

-Clients are growing at a rate of 1.8 per year.

-His profits are $631.42 per client per year.

-His profits are dropping $12.41 per client per year.

I need to find the instantaneous rate of change for his profits this year. For an equation to find his yearly profit I got (631.42(61+1.8x))-(12.41(61(1.8x))) I don't know if this is correct but when I factored it and got its derivative I got just -226.062 and I don't feel that's right.

 

I know you have to find the equation to represent his profits, find the derivative, and the plug in 0 for the year to get the rate of change now but I'm not sure what to do with all the numbers and such.

Answer 1

 

This problem gives a lot of rates.  Let's sort them out because I think you missed a required integration.

 

Let 

  t      = number of years, with current year t=0

  p(t)  = profit per client in year t

  S(t)  = total profit in year t

  c(t)   = number of clients in year t

 

Clients are growing at 1.8 per year:

  c(t) = 61 + 1.8t    (1)     

  

So we know:

   S(t) = c(t)•p(t)      (2)

 

What about profits per client then?  The only information given here is "profits are dropping 12.41 per client per year"  Since this is a rate per year, we have:

 

   d(p(t))

   ————  = -12.41   (profits are dropping $12.41 per client per year)

      dt

 

  and  p(t) = -12.41 ∫ dt

          p(t) = -12.41t + K

 

   p(0) = 631.42  ==> K=631.42

 

           p(t) = -12.41t + 631.42     (3)

 

 

Now plug in (3) and (1) into (2), multiply out and simplify the result, to get:

 

      S(t) = -22.338t² + 379.546t + 38516.62

 

 

Sanity checks:

1.  This year, t=0,  S(0) = 38516.62   (which is 61 clients * 631.42/client)   (ok)

2.  Next year, t=1,  S(1) = 38873.828  (profit increased by $357.208, this is less than 1.8 clients * 631.42 and still a positive amount since profits have only declined 12.41 per client over this first year)

3.  What happens over the long run?  Since the problem states his profits per client are steadily declining, he will eventually not have any profits at all.  That agrees with the -22.338t² term, as that will dominate as t gets larger.