Questions: An insect population decreases at a rate of 12% per year. The insect population on 1st January 2022 is 3.7 million. Calculate the number of complete years it will take for the insect population to first fall below 1 million insects.

Solution Steps

To solve this problem, we need to model the population decrease using exponential decay. We start with an initial population and decrease it by 12% each year until it falls below 1 million. We can use a loop to simulate each year and count the number of years it takes for the population to drop below the threshold.

We start with an initial insect population of \( P_0 = 3.7 \) million on January 1, 2022. The population decreases at a rate of \( r = 0.12 \) (or 12%) per year. We need to determine the number of complete years \( t \) it takes for the population to fall below \( P_{threshold} = 1 \) million.

The population after \( t \) years can be modeled by the formula: \[ P(t) = P_0 \times (1 - r)^t \] Substituting the known values, we have: \[ P(t) = 3.7 \times (1 - 0.12)^t = 3.7 \times 0.88^t \]

We need to find the smallest integer \( t \) such that: \[ P(t) < 1 \] This translates to: \[ 3.7 \times 0.88^t < 1 \] Dividing both sides by 3.7 gives: \[ 0.88^t < \frac{1}{3.7} \approx 0.27027 \]

By iterating through values of \( t \), we find that \( t = 11 \) is the first complete year where the population falls below 1 million. At \( t = 11 \): \[ P(11) = 3.7 \times 0.88^{11} \approx 0.9068 \text{ million} \]

Final Answer