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To solve the question on exponential models and determine the doubling time for the investment, let's break down the information provided.

The investment model is given as:
\[
A(t) = 500(1.0107)^t
\]
where:
- $ A(t) $ represents the investment balance in dollars,
- $ t $ is the number of years,
- The base $ 1.0107 $ represents the growth rate.

### Doubling Time Calculation
To find the doubling time, we want to determine the time $ t $ when the investment amount $ A(t) $ doubles the initial investment. Since the initial amount is $500, doubling means $ A(t) = 1000 $.

1. Set up the equation:
   \[
   1000 = 500 \times (1.0107)^t
   \]

2. Divide both sides by 500:
   \[
   2 = (1.0107)^t
   \]

3. Take the natural logarithm of both sides to solve for $ t $:
   \[
   \ln(2) = t \cdot \ln(1.0107)
   \]

4. Solve for $ t $:
   \[
   t = \frac{\ln(2)}{\ln(1.0107)}
   \]

Calculating this:

\[
t \approx \frac{0.6931}{0.01065} \approx 65.1 \text{ years}
\]

So, the doubling time for the investment is approximately **65.1 years**.

Would you like a further breakdown of the logarithmic steps, or do you have additional questions on this?

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Here are five related questions to explore:

1. How would the doubling time change if the growth rate increased?
2. What is the effect of the initial investment amount on doubling time?
3. Can we generalize a formula for doubling time given any growth rate?
4. How would halving time be calculated for a depreciating investment?
5. How is this model applied in real-world financial calculations?

**Tip:** For exponential models, doubling time can be quickly estimated with the "Rule of 70": divide 70 by the percentage growth rate. This is a handy approximation!

The amount of money in an investment is modeled by the function A(t) = 500(1.0107)^t. The variable A represents the investment balance in dollars, and t the number of years. What is the doubling time for the investment? Round answer to 1 decimal place.

Understand how to calculate the doubling time for an investment modeled by an exponential growth function. This problem covers key mathematical concepts, including exponential growth, logarithmic calculations, and the doubling time formula. Suitable for students in Grade 11-12.

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