Let P(t) be the population of Colorado, in millions, t years after 2000, with P(3) =4.55 and P(9) =5.02
1a) Find a formula for P(t) assuming it is linear. Explain Colorado’s annual population growth using this model.
1b) Find a formula for P(t) assuming it is exponential. Explain Colorado’s population growth using this model.
1c) Explain the difference between how linear and exponential functions progress from one value to the next.
p = b + k t
4.55 = b + k *3
5.02 = b + k * 9
-------------------------subtract
-0.47 = -6 k
k = 0.0783
go back and get b
===============================
p = C e^kt
4.55 = C e^ 3 k
5.02 = C e^ 9 k
ln 4.55 = ln C + 3 k = 1.515
ln 5.02 = ln C + 9 k = 1.613
------------------------------------- subtract
- 6 k = - 0.098
k = 0.0163
so
p = C e^0.0163 t
go back and get C from either data point
straight, curved upward