An investment grows by (y%) annually. If the initial investment is $5,000 and it grows to $6,000 in 2 years, find the annual growth rate (y).

To find the annual growth rate $ y \% $, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{y}{100}\right)^t
\]

Where:
- $ A $ is the amount after time $ t $,
- $ P $ is the initial principal (initial investment),
- $ y $ is the annual growth rate,
- $ t $ is the time in years.

In this problem:
- $ A = 6000 $ (the amount after 2 years),
- $ P = 5000 $ (the initial investment),
- $ t = 2 $ years.

Substituting the values into the formula:

\[
6000 = 5000 \left(1 + \frac{y}{100}\right)^2
\]

Now, divide both sides by 5000 to isolate the growth factor:

\[
\frac{6000}{5000} = \left(1 + \frac{y}{100}\right)^2
\]

Simplifying:

\[
1.2 = \left(1 + \frac{y}{100}\right)^2
\]

Next, take the square root of both sides to solve for $ 1 + \frac{y}{100} $:

\[
\sqrt{1.2} = 1 + \frac{y}{100}
\]

Calculate the square root of 1.2:

\[
1.0954 \approx 1 + \frac{y}{100}
\]

Finally, subtract 1 from both sides and multiply by 100 to solve for $ y $:

\[
\frac{y}{100} \approx 0.0954
\]

\[
y \approx 9.54\%
\]

Thus, the annual growth rate $ y $ is approximately **9.54%**.

Would you like more details on this process, or have any questions? 

Here are some related questions:
1. How would the formula change if the growth occurred continuously instead of annually?
2. What would be the investment's value after 5 years with the calculated growth rate?
3. How would the growth rate change if the investment grew to $7,000 instead of $6,000 in the same period?
4. How would you calculate the time it would take for the investment to double with this growth rate?
5. What impact does compounding frequency (e.g., quarterly, monthly) have on the growth rate?

**Tip:** Always double-check your calculations, especially when dealing with square roots and percentages, to ensure accuracy.

Learn how to calculate the annual growth rate (y%) of an investment from an initial investment of $5,000 growing to $6,000 in 2 years using compound interest. Detailed step-by-step solution provided.

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