A bird species in danger of extinction has a population that is decreasing exponentially (𝐴=𝐴0𝑒𝑘𝑡).

Question

7 years ago the population was at 1640 and today only 1090 of the birds are alive. Once the population drops below 190, the situation will be irreversible. When will this happen? Hint: When you find 𝑘, round it to 6 decimals and use that value to answer the question.

Answer 1

A = A₀e^kt = population t years from 7 years ago

When t = 0, A = 1640. So, A₀ = 1640

A = 1640e^kt

When t = 7, A = 1090. So, 1090 = 1640e^7k

0.664634146 = e^7k

ln(0.664634146) = ln(e^7k) = 7k

So, k = -0.058359792

A = 1640e^-0.058360t

Find t when A = 190:

190 = 1640e^-0.058360t

0.115853658 = e^-0.058360t

ln(0.115853658) = -0.058360t

t ≈ 36.93 years from 7 years ago (or 29.93 years from now)