Question

In 1990 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1997 the population of splake in the lake was estimated to be 3000. Using the Malthusian law for population growth, estimate the population of splake in the lake in the year 2020.

Answer 1

Answer:

The estimation for the population of splake in the lake in the year 2020 is 110,720.

Step-by-step explanation:

The Malthusian law for population growth is given by the following equation:

$P(t) = P(0)e^{rt}$

In which P(0) is the initial population and r is the population growth rate.

In 1990 the Department of Natural Resources released 1000 splake into the lake.

This means that $P(0) = 1000$.

In 1997 the population of splake in the lake was estimated to be 3000.

1997 is 7 years after 1990. This means that $P(7) = 3000$

Replacing this into the equation, we find the value of r.

$P(t) = P(0)e^{rt}$

$3000 = 1000e^{7r}$

$e^{7r} = 3$

Applying ln to both sides

$\ln{e^{7r}} = \ln{3}$

$7r = \ln{3}$

$r = \frac{\ln{3}}{7}$

$r = 0.1569$

So

$P(t) = 1000e^{0.1569t}$

Using the Malthusian law for population growth, estimate the population of splake in the lake in the year 2020.

2020 is 30 years after 1990. So this is P(30).

$P(t) = 1000e^{0.1569t} = 1000e^{0.1569*30} = 110,720$

The estimation for the population of splake in the lake in the year 2020 is 110,720.