Math Problem- Population

Question

A bird species in danger of extinction has a population that is decreasing exponentially (A = A₀e^kt). Five years ago, the population was at 1400 and today only 1000 of the birds are alive. Once the population drops below 100, the situation will be irreversible. When will this happen?

Answer 1

The key to all of the parts is using natural logarithms to isolate and solve for the kt exponent terms. A₀ is the initial population before the decrease.

First, calculate the decay constant k

A = A₀e^kt

A = e^kt

A₀

ln(A/A₀) = ln(e^kt)

ln(A/A₀) = kt

k = ln(A/A₀)

t

k = ln(1000/1400)

5

k = −0.067

Use k to solve for the number of years for population to fall below 100. I will use 99 as the population.

A = A₀e^kt

A = e^kt

A₀

ln(A/A₀) = ln(e^kt)

ln(A/A₀) = kt

t = ln(A/A₀)

k

t = ln(99/1400)

-0.067

t = 39.5 yrs