https://file.aiquickdraw.com/mathbot/file/17307672586653qzmzjic.jpg

Let's analyze the information provided and solve the problem step-by-step.

The assets \( A(x) \) for a financial firm can be modeled by the function:
\[
A(x) = 342e^{0.057x}
\]
where \( x = 7 \) corresponds to the year 2007. We need to find the value of the assets for the years 2012, 2014, and 2018.

### Step 1: Determine the \( x \)-values for each year
Since \( x = 7 \) corresponds to 2007, we calculate \( x \) for each target year:
1. **2012**: \( x = 7 + (2012 - 2007) = 7 + 5 = 12 \)
2. **2014**: \( x = 7 + (2014 - 2007) = 7 + 7 = 14 \)
3. **2018**: \( x = 7 + (2018 - 2007) = 7 + 11 = 18 \)

### Step 2: Substitute each \( x \)-value into the function
1. **For 2012 ( \( x = 12 \) ):**
   \[
   A(12) = 342e^{0.057 \times 12}
   \]
2. **For 2014 ( \( x = 14 \) ):**
   \[
   A(14) = 342e^{0.057 \times 14}
   \]
3. **For 2018 ( \( x = 18 \) ):**
   \[
   A(18) = 342e^{0.057 \times 18}
   \]

Let's calculate each expression individually to find the approximate values of the assets for each year. I’ll compute these for you:

#### Calculations
1. **2012 ( \( x = 12 \) ):**
   \[
   A(12) \approx 342 \times e^{0.684} \approx 677.8 \, \text{billion}
   \]

2. **2014 ( \( x = 14 \) ):**
   \[
   A(14) \approx 342 \times e^{0.798} \approx 801.5 \, \text{billion}
   \]

3. **2018 ( \( x = 18 \) ):**
   \[
   A(18) \approx 342 \times e^{1.026} \approx 1094.6 \, \text{billion}
   \]

### Summary of Results
- **2012**: Approximately **677.8 billion dollars**
- **2014**: Approximately **801.5 billion dollars**
- **2018**: Approximately **1094.6 billion dollars**

These answers should match the options provided in the question, rounded to the nearest tenth as needed. Let me know if you'd like more details on any part of the solution, or if you have any further questions!

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### Related Questions to Explore
1. How does the growth rate parameter \( 0.057 \) affect the asset growth over time?
2. What would the assets be in the year 2025 using this model?
3. How would a change in the base year (e.g., setting 2010 as \( x = 0 \)) affect the function?
4. Can we predict future values beyond 2018 with reasonable accuracy using exponential models?
5. How sensitive is the function to small changes in \( x \)?

### Tip
When working with exponential functions, understanding the base year (starting point) is crucial, as it allows us to accurately determine the value of \( x \) for any target year.

The assets (in billions of dollars) for a financial firm can be approximated by the function A(x) = 342e^(0.057x), where x = 7 corresponds to the year 2007. Find the assets in each of the following years: (a) 2012, (b) 2014, (c) 2018.

Learn how to calculate the assets of a financial firm for the years 2012, 2014, and 2018 using the exponential growth model A(x) = 342e^(0.057x). This problem involves key concepts in exponential functions and algebra, suitable for high school students in Grades 10-12.

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