Hi i am really struggling to answer this calculus question

Question

The manager of the Many Facets jewelry store models total sales by the function S(t)=(2000t)/(4+0.1t) where t is the time (years) since the year 2005 and SS is measured in thousands of dollars.

 

At what rate (in dollars per year) were sales changing in the year 2009?

 

 

Answer 1

S(t) = 2000t / (4+0.1t);   t=0 at year 2005

 

You need rate of change, that implies taking a derivative:

 

Let's write S(t) as this:  S(t) = 20000 * (t/(40+t))

 

Using the quotient rule: 

S'(t) = 20000 * (40 / (40+t)²)

S'(t) =  800000/(40+t)² 

 

 

2009 is represented by t=4, so just plug in t=4 into S'(t)

S'(4) = 800000 / (40+4)² = _________________ thousands of dollars per year

 

If needed, multiply the answer you get by 1000 to get dollars per year.

 

Answer 2

Take t=1 for 2005, Calculate S(1).

2005: t=1, S(1)=(2000(1)/(4+0.1 *1)

2009  t=5  S(5)=(2000*5)/(4+0.1*5)

S(5)-S(1) is interest. Next, find interest rate as a percentage from S(5)-S(1) as (S(5)-S(1))/S(1).This is rat for 5 years. Divide it by 5 and obtain annual interest rate.