Question

The population of a small town is modeled by an exponential function of the form p t( ) = abt , where t represents the number of years since 2010. The population of the town was recorded as 425 in 2010 and 612 in 2012. Based on the data for the years 2010 and 2012, what is the value of b in the model?

Answer 1

Answer:

$\sqrt{\frac{36}{25}} = b= \frac{6}{5}$

So then the model would be given by:

$p(t) = 425 (\frac{6}{5})^t$

Step-by-step explanation:

For this case we assume the following model:

$p(t) = ab^t$

And we know that p represent the population , and t the years since 2010. We have two initial conditions given:

$t =0 , P(0) = 425$

Since for 2010 is the staarting point, t=0

And using this condition we got:

$425 = a b^t = a$

And from the other condition:

$t =2 , P(2) = 612$

$612=425b^2$

If we divide both sides by 425 we got:

$\frac{612}{425}=\frac{36}{25}= b^2$

And if we apply square root on both sides we got:

$\sqrt{\frac{36}{25}} = b= \frac{6}{5}$

So then the model would be given by:

$p(t) = 425 (\frac{6}{5})^t$