If the initial population of rats was 20 and grew to 25 after the second year, which of the following functions best models the population of rats P with respect to the number of years t if the population growth of rats is considered to be exponential?
If the initial population of rats was 20 and grew to 25 after the second year, which of the following functions best models the population of rats P with respect to the number of years t if the population growth of rats is considered to be exponential?
Explanation
Since the population growth is exponential, the population P can be modeled as:
P = P0 x (1 + r)^t
where P0 is the initial population, r is the growth rate, and t is the time (in years).
Given that the initial population P0 = 20 and the population grew to 25 after 2 years, we can find the growth rate r:
25 = 20 x (1 + r)^2
Dividing both sides by 20:
1.25 = (1 + r)^2
Taking the square root:
1 + r = √1.25 = 1.125
Subtracting 1 from both sides:
r = 0.125
Now we can write the exponential growth function:
P = 20 x (1 + 0.125)^t
P = 20 x (1.25)^t