4 years ago, you invested $15,500. In 5 years, you expect to have $38,916. If you expect to earn the same annual return after 5 years from today as the annual return implied from the past and expected values given in the problem, then in how many years from today do you expect to have $49,795? Round final answer to 2 decimal places.

We are tasked with determining in how many years from today you will expect to have $49,795 based on the investment growth.

### Step 1: Find the annual growth rate (implied return)
We know the investment grows from $15,500 to $38,916 in 9 years (4 years ago to 5 years from today). We can use the compound interest formula to find the implied annual growth rate:

\[
A = P(1 + r)^t
\]
where:
- $ A $ = final amount ($38,916),
- $ P $ = initial investment ($15,500),
- $ r $ = annual growth rate,
- $ t $ = time in years (9 years in this case).

Rearranging the formula to solve for $ r $:

\[
r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1
\]

Substitute the given values:

\[
r = \left( \frac{38,916}{15,500} \right)^{\frac{1}{9}} - 1
\]

### Step 2: Calculate the annual growth rate
Let me calculate this.The annual growth rate $ r $ is approximately 10.77%.

### Step 3: Determine the time required to reach $49,795
Now, we can use the same compound interest formula to determine the time $ t' $ it will take for the investment to grow from $38,916 to $49,795 with this annual growth rate:

\[
A' = P'(1 + r)^{t'}
\]

Where:
- $ A' = 49,795 $,
- $ P' = 38,916 $,
- $ r = 0.1077 $.

Rearranging to solve for $ t' $:

\[
t' = \frac{\ln(A' / P')}{\ln(1 + r)}
\]

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later. However, you can try calculating the result using the formula provided earlier:

\[
t' = \frac{\ln\left(\frac{49,795}{38,916}\right)}{\ln(1 + 0.1077)}
\]

This will give you the time $ t' $ in years.

If you have more questions, feel free to ask! Would you like help with any other steps or a similar problem?

Here are five related questions you might explore:
1. How does compound interest differ from simple interest?
2. What factors influence the annual growth rate of an investment?
3. How can you calculate the present value of future cash flows?
4. What is the effect of different compounding periods (monthly, quarterly) on interest rates?
5. How do inflation rates

Learn how to calculate the time it takes to grow an investment from $38,916 to $49,795 using compound interest. This problem demonstrates key concepts such as exponential growth, the compound interest formula, and logarithmic calculations, suitable for students in Grades 10-12.

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