The population​ (in thousands) of people of a city is approximated by the function Upper P left parenthesis t right parenthesis equals 1200 left parenthesis 2 right parenthesis Superscript 0.1009 t​, where t is the number of years since 2011.
a. Find the population of this group in 2019.
b. Predict the population in 2028.


The population function is given by:

\[
P(t) = 1200 \times 2^{0.1009t}
\]

where \( t \) is the number of years since 2011, and \( P(t) \) gives the population in thousands.

### (a) Finding the population in 2019:

To find the population in 2019, we first determine the value of \( t \) for 2019. Since \( t \) is the number of years since 2011:

\[
t = 2019 - 2011 = 8
\]

Now, substitute \( t = 8 \) into the population function:

\[
P(8) = 1200 \times 2^{0.1009 \times 8}
\]

First, calculate the exponent \( 0.1009 \times 8 = 0.8072 \). Then calculate \( 2^{0.8072} \).

\[
P(8) = 1200 \times 2^{0.8072}
\]

Finally, multiply by 1200 to get the population in 2019.

### (b) Predicting the population in 2028:

Similarly, for 2028:

\[
t = 2028 - 2011 = 17
\]

Substitute \( t = 17 \) into the population function:

\[
P(17) = 1200 \times 2^{0.1009 \times 17}
\]

First, calculate \( 0.1009 \times 17 = 1.7153 \). Then calculate \( 2^{1.7153} \).

\[
P(17) = 1200 \times 2^{1.7153}
\]

Now, multiply by 1200 to get the predicted population in 2028.

I'll compute these values now.### Results:
(a) The population in 2019 is approximately **2,099.77 thousand** (or 2.1 million).

(b) The predicted population in 2028 is approximately **3,940.38 thousand** (or 3.94 million).

Would you like further details or have any questions?

Here are five related questions to expand on this topic:
1. How does the growth rate affect long-term population predictions?
2. What would the population have been in 2011, based on this model?
3. How can exponential growth models be adjusted for real-world factors like migration or natural disasters?
4. Can you calculate the year when the population will reach 5 million?
5. How does the base of the exponential function \( 2 \) affect the speed of population growth?

**Tip**: Exponential functions grow faster over time, so even small increases in the exponent can lead to significant changes in predictions.

The population (in thousands) of people of a city is approximated by the function P(t) = 1200(2)^0.1009t, where t is the number of years since 2011. a. Find the population of this group in 2019. b. Predict the population in 2028.

Solve an exponential growth problem where the population of a city is modeled by P(t) = 1200(2)^0.1009t. Find the population in 2019 and predict it for 2028, learning about key concepts like exponential growth and population modeling. Suitable for Grades 10-12.

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