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

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Step-by-step explanation:

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Answer 2

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=(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