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).

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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.