Question

solve the given problem related to population growth.During the first decade of this century the population of a certain city grew exponentially. The population of the city was 143,230 in 2000 and 217,325 in 2010. Find the exponential growth function that models the population growth of the city. Use t=0 to represent 2000 t =10 to represent 2010. Round k to five decimal places.N (t)=Use your exponential growth function to predict the population of the city in 2016. Round to the nearest thousand

Answer 1

Explanation:

The exponential growth function has the form

$N\left(t\right)=N_0e^{kt}$

Where N0 is the population for t = 0, k is a constant, and t is the number of years.

When t = 0, the population is 143,230, so

$N\left(t\right)=143230e^{kt}$

To find k, we will use the given information that when t = 10, the population N(t) = 217,325. So, by replacing these values and solving for k, we get:

$\begin{gathered} 217325=143230e^{k\left(10\right)} \\ \frac{217325}{143230}=e^{10k} \\ 1.52=e^{10k} \\ \ln1.52=\ln e^{10k} \\ 0.4169=10k \\ \frac{0.4169}{10}=k \\ 0.04169=k \end{gathered}$

Therefore, the equation that models the population growth is

$N\left(t\right)=143230e^{0.04169t}$

Finally, to predict the population of the city in 2016, we need to replace t = 16, so

$\begin{gathered} N\left(16\right)=143230e^{0.04169\left(16\right)} \\ N\left(16\right)=279,079.116 \end{gathered}$

Answer:

So, the answers are

$N\left(t\right)=143230e^{0.04169t}$

Population in 2016: 279,079.116