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

The exponential growth function that models the population growth of the city can be found using the formula N(t) = P * e^(kt). By plugging in the given population values for 2000 and 2010, we can solve for the growth rate k. Using the exponential growth function, we can predict the population of the city in 2016.

Step-by-step explanation:

To find the exponential growth function that models the population growth of the city, we can use the formula N(t) = P * e^(kt), where N(t) is the population at time t, P is the initial population, e is the base of the natural logarithm, and k is the growth rate.

Given that the population was 143,230 in 2000 and 217,325 in 2010, we can plug these values into the formula and solve for k. Substituting t = 0 and N(t) = 143,230, we get 143,230 = P * e^(k(0)), which simplifies to P = 143,230.

Substituting t = 10 and N(t) = 217,325, we get 217,325 = 143,230 * e^(k(10)), which simplifies to e^(k(10)) = 217,325/143,230.

Now we can find k by taking the natural logarithm of both sides of the equation. Solving for k gives us k ≈ ln(217,325/143,230)/10.

Once we have the value of k, we can use the exponential growth function to predict the population in 2016. Substituting t = 16 into the exponential growth function N(t) = P * e^(kt), using the initial population P = 143,230 and the value of k we found, we can calculate the population to the nearest thousand.

Answer 2

Step-by-step 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)) \\ (217325)/(143230)=e^(10k) \\ 1.52=e^(10k) \\ \ln1.52=\ln e^(10k) \\ 0.4169=10k \\ (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