Question

Suppose that the population of a city was about 370 thousand in 2000 and had been growing by about 8.5% per year.(a) Write an explicit formula for the population of the city t years after 2000 (i.e. t= 0 in 2000), where Pt is measured in thousands of people.Pt = (b) If this trend continues, what will the city's population be in 2016? Round your answer to the nearest whole number. thousand people(c) When does this model predict city's population to exceed 600 thousand? Give your answer as a calendar year (ex: 2000).During the year

Answer 1

The growth is an exponential growth

(a)

`\begin{gathered} \text{ The explicit formula for the population of the city can be represented by } \\ p(t)=p(1+r)^t \\ \text{Where p=initial population} \\ r=\text{growth rate} \\ t=\text{ number of years} \end{gathered}`

For t years after 2000, at 8.5% growth rate per year , we have

`\begin{gathered} p=2000,\text{ r=8.5\%=}(8.5)/(100)=0.085\text{ } \\ p(t)=2000(1+0.085)^t \\ p(t)=2000(1.085)^t \end{gathered}`

(b) If this trend continues, what will the city's population be in 2016?

from year 2000 to 2016, we have 16 years. Therefore, t=16

put t=16 into p(t) above

we have

`\begin{gathered} p(t)=2000(1.085)^t \\ p(16)=2000(1.085)^(16) \\ p(16)=2000(3.68872) \\ p(16)=7377.44_{} \\ p(16)=7377(\text{ nearest whole number)} \end{gathered}`

The population of the city in 2016 will be 7377 (to nearest whole number)

When does this model predict city's population to exceed 600 thousand?

This can be interpreted as the year when p(t) is greater than 600,000

`\begin{gathered} \text{ that is solve for t when p(t)>600,000} \\ 2000(1.085)^t>600,000 \\ \text{divide both sides by 2000} \\ 1.085^t>300 \\ \text{take logarithm of both sides} \\ \log (1.085)^t>\log 300 \\ t\log 1.085>2.47712 \\ t>(2.47712)/(\log 1.085) \\ t>69.916 \\ t>70\text{ (nearest whole number)} \\ \end{gathered}`

This implies that from 70 years from year 2000, the city's population exceed 600 thousand.

In calendar, this will be the year 2070.

Hence, in year 2070 the city's population will exceed 600 thousand.