Question

Suppose the population of a town was 40,000 on January 1, 2010 and was 50,000 on January 1, 2015. Let P(t) be the population of the town in thousands of people t years after January 1, 2010.
6 (a) Build an exponential model (in the form P(t) = a bt ) that relates P(t) and t. Round the value of b to 5 significant figures.

a = ?

b = ?

Answer 1

Answer:

Given,

The initial population ( on 2010 ) = 40,000,

Let r be the rate of increasing population per year,

Thus, the function that shows the population after t years,

$P(x)=40000(1+r)^t$

And, the population after 5 years ( on 2015 ) is,

$P(5)=40000(1+r)^{5}$

According to the question,

P(5) = 50,000,

$\implies 40000(1+r)^5=50000$

$(1+r)^5=\frac{50000}{40000}=1.25$

$r + 1= 1.04563955259$

$\implies r = 0.04653955259\approx 0.04654$

So, the population is increasing the with rate of 0.04654,

And, the population after t years would be,

$P(t)=40000(1+0.04654)^t$

$\implies 40000(1.04654)^t$

Since, the exponential function is,

$f(x) = ab^x$

Hence, by comparing,

a = 40000,

b = 1.04654