The number of people living in the planned seaside community of Sunnyday Florida is modeled by the functionP(t)=3100+8.5t+0.1t^2

Question

The number of people living in the planned seaside community of Sunnyday Florida is modeled by the function P(t)=3100+8.5t+0.1t^2 where t =0 represents 1990 when the community was incorporated. a. At what rate is the number of people living in Sunnyday changing in 1995? b. What is the average rate of change in the number living of people living in Sunnyday between 1995 and 2001?

Answer 1

To obtain the rate of change at a point, you can take the derivative of the function P(t)

P'(t) = 0+8.5+0.1*2*t = 8.5 + 0.2t 

Plug in 5 for t as (1995-1990 = 5)

 

P'(5) = 8.5 + 0.2*5 is the rate at which the number of people living in Sunnyday are changing in 1995.

 

To get average rate of change between 1995 and 2001 though you need to do the following

Calculate the number of people living in Sunnday in 2001 = P(11) as it is 11 years from 1990

Calculate the number of people living in Sunnday in 1995 = P(5) as it is 5 years from 1990

Calculate P(11)-P(5) that will give you the change in the people that happened between 1995 and 2001

Then divide the result with 6 to know the average rate of change i.e (P(11) - P(5))/6