Question

A bird species in danger of extinction has a population that is decreasing exponentially. Seven years ago the

population was at 1800 and today only 1000 of the birds are
alive. Once the population drops below 200, the situation will
be irreversible. How many years from now will this happen?

Answer 1

Using the exponential decay formula, the bird population will drop below 200 in approximately 13.59 years from now.

The student's question involves a bird species with a population decreasing exponentially. We are given the past and current sizes of the population and asked to determine how many more years it will take for the population to drop below 200. To solve this, we use the formula for exponential decay which is N(t) = N0 * e^(-rt), where N(t) is the population at time t, N0 is the initial population, e is the base of the natural logarithm, and r is the rate of decay.

Given:

N0 (7 years ago) = 1800

N(t) (now) = 1000

Time interval (∆t) = 7 years

We first calculate the decay rate using:

1000 = 1800 * e^(-7r)

Solving for r, we get:

r ≈ 0.1225 per year

Now, we want to find the time t when N(t) reaches 200:

200 = 1000 * e^(-rt)

Solving for t we get:

t ≈ 13.59 years from now

The population will drop below 200 in approximately 13.59 years from today.