Step-by-step explanation:
Population 1:
Initial population (P0) = 20
Final population (P) = 160
Time (t) = 7 years
We can use the exponential growth model: P(t) = P0ekt
160 = 20e^(7k)
To find the growth rate (k), we can divide both sides by 20:
8 = e^(7k)
Take the natural logarithm of both sides:
ln(8) = 7k
k = ln(8) / 7 ≈ 0.277
So, the growth model for Population 1 is: P(t) = 20e^(0.277t)
Population 2:
Initial population (P0) = 40
Growth rate (k) = half of Population 1's growth rate = 0.277 / 2 = 0.1385
We want to find the time (t) when the two populations are equal in size.
Let's set up an equation using the growth models:
20e^(0.277t) = 40e^(0.1385t)
Divide both sides by 20:
e^(0.277t) = 2e^(0.1385t)
Take the natural logarithm of both sides:
0.277t = ln(2) + 0.1385t
Subtract 0.1385t from both sides:
0.1385t = ln(2)
Divide both sides by 0.1385:
t ≈ 5.03 years
Therefore, the two populations will become equal in size after approximately 5.03 years.
