Tim is doing a research project on a species of endangered turtle, and he found out that the species' population is rebounding as a result of conservation efforts. This function models the expected population t years from now:
P(t)=a(b)t, where a and b are constants
This table shows two values of the function:
t P(t)
2 12,100
3 13,310
Find an expression for P(t). Write your answer in the form P(t)=a(b)t, where a and b are integers or decimals. Do not round.
Using the given values from the table, we can create two equations:
P(2) = a(b)^2 = 12,100
P(3) = a(b)^3 = 13,310
Dividing the second equation by the first equation gives:
P(3)/P(2) = (a(b)^3)/(a(b)^2) = (a(b)^3)(1/a(b)^2) = b
Therefore, we can conclude that b = P(3)/P(2) = 13,310/12,100.
Substituting this value of b into the first equation:
12,100 = a(b)^2 = a(13,310/12,100)^2
Simplifying:
12,100 = a(1.100)^2 = a(1.21)
Therefore, a = 12,100/1.21 = 10,000.
So the expression for P(t) is:
P(t) = 10,000(13,310/12,100)^t