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Given the exponential function $ A(t) = 4200(1.054)^t $:

1. **The initial amount invested is**:
   - The initial amount is the value of the investment when $ t = 0 $.
   - Therefore, $ A(0) = 4200(1.054)^0 = 4200 $.
   - The initial amount invested is **$4200**.

2. **The multiplier is**:
   - The multiplier is the base of the exponential function.
   - Therefore, the multiplier is **1.054**.

3. **The percent growth rate is**:
   - The percent growth rate can be derived from the multiplier.
   - The formula for the growth rate $ r $ is: $ 1 + r = 1.054 $.
   - Thus, $ r = 0.054 $ or 5.4%.
   - The percent growth rate is **5.4%**.

4. **The value of the investment after 10 years is**:
   - Substitute $ t = 10 $ into the function $ A(t) $.
   - $ A(10) = 4200(1.054)^{10} $.
   - Calculate $ A(10) $:
     \[
     A(10) = 4200 \times (1.054)^{10} \approx 4200 \times 1.685 \approx 7077
     \]
   - The value of the investment after 10 years is approximately **$7077.00**.

5. **The doubling time of the investment is**:
   - The doubling time $ T $ can be found using the rule $ 2 = (1.054)^T $.
   - Taking the natural logarithm of both sides: $ \ln(2) = T \ln(1.054) $.
   - Solve for $ T $:
     \[
     T = \frac{\ln(2)}{\ln(1.054)} \approx \frac{0.693}{0.0526} \approx 13.18
     \]
   - The doubling time of the investment is approximately **13.18 years**.

Would you like more details or have any questions?

Here are 5 additional questions you might ask:

1. How do you calculate the future value of an investment with compound interest?
2. What is the difference between simple interest and compound interest?
3. How do logarithms help in solving exponential equations?
4. What factors can affect the growth rate of an investment?
5. Can you explain the rule of 72 for estimating doubling time?

**Tip:** When dealing with exponential growth, understanding the base of the exponential function and its relationship to the growth rate is crucial for making accurate predictions.

Explore the concept of exponential growth using an investment example. Learn how to calculate initial investment, growth rate, future value after a period, and doubling time. Suitable for advanced high school students and above.

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