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?
- P=20(5)t
- P=5t^2+20
- P=20(1.25)^t
- None of these
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
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