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 142,390 in 2000 and 212,155 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, and so on. (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.

To model the population growth using an exponential growth function, we use the formula:

\[
N(t) = N_0 e^{kt}
\]

Where:
- \(N(t)\) is the population at time \(t\),
- \(N_0\) is the initial population (at \(t = 0\)),
- \(k\) is the growth rate,
- \(t\) is the time in years after 2000.

### Step 1: Set up the initial values.
From the problem:
- The population in 2000 (at \(t = 0\)) is \(N_0 = 142,390\).
- The population in 2010 (at \(t = 10\)) is \(N(10) = 212,155\).

### Step 2: Use the population data from 2010 to solve for \(k\).
We know:
\[
N(10) = N_0 e^{10k}
\]
Substitute the values:
\[
212,155 = 142,390 e^{10k}
\]

Solve for \(e^{10k}\):
\[
\frac{212,155}{142,390} = e^{10k}
\]

\[
1.489749 = e^{10k}
\]

Take the natural logarithm of both sides:
\[
\ln(1.489749) = 10k
\]

\[
0.398776 = 10k
\]

Solve for \(k\):
\[
k = \frac{0.398776}{10} = 0.03988
\]

So, the exponential growth function is:
\[
N(t) = 142,390 e^{0.03988t}
\]

### Step 3: Predict the population in 2016.
To predict the population in 2016, we use \(t = 16\) (since 2016 is 16 years after 2000).

Substitute \(t = 16\) into the growth function:
\[
N(16) = 142,390 e^{0.03988 \times 16}
\]

First, calculate the exponent:
\[
0.03988 \times 16 = 0.63808
\]

Now, calculate \(e^{0.63808}\):
\[
e^{0.63808} \approx 1.893
\]

Now, multiply by the initial population:
\[
N(16) = 142,390 \times 1.893 \approx 269,632
\]

### Final Answer:
The population of the city in 2016 is approximately **269,000** (rounded to the nearest thousand).

---

Would you like more details or clarifications?

Here are some related questions to expand on the topic:
1. How do you calculate population growth if the growth rate changes over time?
2. What is the difference between exponential and logistic growth models?
3. How would the model change if we considered factors that limit population growth?
4. How can we estimate the future population if the population decreases over time?
5. What are some real-life examples where exponential growth applies?

**Tip:** Exponential models are useful for populations and investments, but they assume unlimited growth, which may not be realistic long-term.

During the first decade of this century, the population of a certain city grew exponentially. The population of the city was 142,390 in 2000 and 212,155 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, and so on. (Round k to five decimal places.) Use your exponential growth function to predict the population of the city in 2016. Round to the nearest thousand.

Solve an exponential growth population problem. The city's population in 2000 was 142,390, and in 2010 it was 212,155. Learn how to model this growth with an exponential function and predict the population for 2016. This problem involves exponential growth concepts and uses the formula N(t) = N_0 e^{kt}, suitable for Grades 11-12.

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