Question

A town has a population of 1600 people at time ????=0. In each of the following cases, write a formula for the population P, of the town as a function of year ????.

Answer 1

Answer:

a) $P(t) =1600 +50t$

b) $P(t) = 1600 (1.05)^t$

Step-by-step explanation:

Assuming the complete question : "A town has a population of 1000 people at time t = 0. In each of the following cases, write a formula for the population, P, of the town as a function of year t. (a) The population increases by 50 people a year. (b) The population increases by 5% a year."

Part a

For this case we can use a linear model in order to estimate the population size since we have a fixed increase each year. So our model would be given by:

$P(t) = P_o + b t$

Where $b =50$ on this case since represent the increase per year of the slope for the linear model. And the initial amount is $P_o = 1600$, so then the model is:

$P(t) =1600 +50t$

Part b

For this case we have a rate of increase and when we have this the lineal model is not the most appropiate. So then we can use an exponential model given by:

$P(t) =P_o b^{t}$

Where $P_o = 1600$ represent the initial population and for this case b is the rate of increase $b = 1+0.05 = 1.05$ since each year we have an increase of 5% and t is the time. So then the model is given by:

$P(t) = 1600 (1.05)^t$