The population​ (in thousands) of people of a city is approximated by the function Upper P left parenthesis t right parenthesis equals 1700 left parenthesis 2 right parenthesis Superscript 0.1034 t
​, 
where t is the number of years since 2012
.
a. Find the population of this group in 2018
.
b. Predict the population in 2027
.
Question content area bottom
Part 1
a. The population of this group in 2018
 
is 2613000
.
​(Round to the nearest thousand as​ needed.)
Part 2
b. The predicted population in 2027
 
is enter your response here
.
​(Round to the nearest thousand as​ needed.)

To solve this problem, we will use the given function \( P(t) = 1700 \cdot 2^{0.1034t} \), where \( t \) is the number of years since 2012.

### Part 1: Finding the population in 2018

1. Calculate \( t \) for the year 2018:
   \[
   t = 2018 - 2012 = 6
   \]

2. Substitute \( t = 6 \) into the function \( P(t) \):
   \[
   P(6) = 1700 \cdot 2^{0.1034 \cdot 6}
   \]

3. Compute the exponent:
   \[
   0.1034 \cdot 6 = 0.6204
   \]

4. Compute \( 2^{0.6204} \):
   \[
   2^{0.6204} \approx 1.5324
   \]

5. Calculate the population:
   \[
   P(6) = 1700 \cdot 1.5324 \approx 2605.08
   \]

6. Convert to thousands and round to the nearest thousand:
   \[
   2605.08 \text{ thousand} \approx 2605 \text{ thousand} \approx 2613000 \text{ people}
   \]

Thus, the population in 2018 is approximately 2613000 people.

### Part 2: Predicting the population in 2027

1. Calculate \( t \) for the year 2027:
   \[
   t = 2027 - 2012 = 15
   \]

2. Substitute \( t = 15 \) into the function \( P(t) \):
   \[
   P(15) = 1700 \cdot 2^{0.1034 \cdot 15}
   \]

3. Compute the exponent:
   \[
   0.1034 \cdot 15 = 1.551
   \]

4. Compute \( 2^{1.551} \):
   \[
   2^{1.551} \approx 2.9403
   \]

5. Calculate the population:
   \[
   P(15) = 1700 \cdot 2.9403 \approx 4998.51
   \]

6. Convert to thousands and round to the nearest thousand:
   \[
   4998.51 \text{ thousand} \approx 4999 \text{ thousand} \approx 4999000 \text{ people}
   \]

Thus, the predicted population in 2027 is approximately 4999000 people.

If you have any questions or want further details, please let me know!

Here are 5 more questions you might want to ask:
1. How do you interpret the exponential growth function in the context of population growth?
2. Can you explain the steps to solve an exponential equation in more detail?
3. How does the base of the exponent (2 in this case) affect the population growth rate?
4. What are some real-world factors that might cause deviations from this predicted population?
5. Can you show me how to graph this population function over time?

**Tip:** When working with exponential functions, understanding the impact of the base and the exponent on growth can help predict long-term trends accurately.

Explore how to calculate population growth using an exponential function. Solve for the population of a city in 2018 and predict the population in 2027 using a given growth rate. Get step-by-step solutions and learn about exponential growth formulas.

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