An animal population numbered 20 at the start of a study (time 0) and after seven years numbered 160. A similar population of animals numbered 40 at the start of the study, and has half the growth rate as the other population. Assuming each population growth follows an exponential model of the form P(t) = Pekt, when do the two populations become equal in size?

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midorimondol midorimondol · Answer 1 · 2024-06-17T06:22:04-04:00

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.