A population of cockroaches that breeds throughout the year started with 13 individuals in our 4th floor cafeteria. If the instantaneous annual rate of increase for this species is 0.67, how many cockroaches can we expect to see in five years, assuming they always find food and space to live?

Question

24-06-2024
Biology

High School

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A population of cockroaches that breeds throughout the year started with 13 individuals in our 4th floor cafeteria. If the instantaneous annual rate of increase for this species is 0.67, how many cockroaches can we expect to see in five years, assuming they always find food and space to live?

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halakukhan1315 halakukhan1315 · Answer 1 · 2024-06-23T18:48:37-04:00

After five years, you can expect to see approximately 370 cockroaches in the cafeteria.

The formula to calculate the population size after a certain time using the instantaneous annual rate of increase is given by:

[tex]N_t[/tex] = [tex]N_0[/tex] [tex]\times e^{rt}[/tex]

where:

- [tex]N_t[/tex] is the population size after time t,

- [tex](\(N_0\))[/tex] is the initial population size,

- r is the instantaneous annual rate of increase, and

- t is the time in years.

Given that the initial population size [tex](\(N_0\))[/tex] is 13, the instantaneous annual rate of increase (r) is 0.67, and the time (t) is 5 years, we can plug these values into the formula to find [tex]N_t[/tex]:

[tex]N_t[/tex] = [tex]13 \times e^{0.67 \times 5}[/tex]

[tex]N_t[/tex] = 13 [tex]\times[/tex] e[tex]^{3.35}[/tex]

[tex]N_t[/tex] ≈ 13 [tex]\times[/tex] 28.509

[tex]N_t[/tex] ≈ 370.617

So, after five years, the population of cockroaches is approximately 371. However, since we are looking for a whole number (cockroaches can't exist in fractions), we would round down to the nearest whole number, which is 370.