The population of a city follows the logistic growth and is limited to 100 000. If the population in 1995 was 50 000 and in 2000 was 60 000, what is the populat

Question

The population of a city follows the logistic growth and is limited to 100 000. If the population in 1995 was 50 000 and in 2000 was 60 000, what is the population in 2005? Give your answer to the nearest hundred.

Answer 1

I had to look up logistic growth on wikipedia.

Here's the equation:

f(x) = L / [1+e^-k(x-x₀)]

Where L = 100,000, the maximum limit.

x₀=1995 is the x value at the middle of the growth range (50,000).

k is a constant that sets the steepness of the curve.

 

f(x) = 100000 / {1+exp[-k(x-1995)]}

f(1995) = 100000 / {1+exp[-k(1995-1995)]} = 50000

100000 / 50000 = 1+exp[-k(0)]

2 = 1+1 = 2

This result just verifies that we've interpreted the equation correctly.  f(1995) =  50000 is indeed the middle of the growth curve.

 

Now we can use the f(2000) point to find k.

f(2000) = 100000 / {1+exp[-k(2000-1995)]} = 60000
100000 / 60000 = 1+exp[-k(5)]

1.6666-1 = exp(-5k)

0.6666 = exp(-5k)

ln(0.6666) = -5k

-0.405 = -5k

0.081 = k

 

f(x) = 100000 / {1+exp[-(0.081)(x-1995)]}

f(2005) = 100000 / {1+exp[-(0.081)(2005-1995)]}

f(2005) = 100000 / {1+exp[-(0.081)(10)]}

f(2005) = 100000 / {1+exp(-0.810)

f(2005) = 100000 / (1+0.444)

f(2005) = 100000 / 1.444 = 69231

f(2005) ≈ 69,200