5An investment of $8,000 grows to $10,000 in 3 years. If the growth is compounded annually, what is the annual growth rate ( y )?

To find the annual growth rate $ y \% $ when an investment grows from $8,000 to $10,000 in 3 years with annual compounding, we can use the compound interest formula:

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

Where:
- $ A = 10,000 $ (the amount after 3 years),
- $ P = 8,000 $ (the initial investment),
- $ t = 3 $ years.

Substituting the values into the formula:

\[
10,000 = 8,000 \left(1 + \frac{y}{100}\right)^3
\]

Now, divide both sides by 8,000 to isolate the growth factor:

\[
\frac{10,000}{8,000} = \left(1 + \frac{y}{100}\right)^3
\]

Simplifying:

\[
1.25 = \left(1 + \frac{y}{100}\right)^3
\]

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

\[
\sqrt[3]{1.25} = 1 + \frac{y}{100}
\]

Calculate the cube root of 1.25:

\[
1.0772 \approx 1 + \frac{y}{100}
\]

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

\[
\frac{y}{100} \approx 0.0772
\]

\[
y \approx 7.72\%
\]

Thus, the annual growth rate $ y $ is approximately **7.72%**.

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

Here are some related questions:
1. What would be the future value of the investment after 5 years with this growth rate?
2. How would the growth rate change if the investment period were extended to 5 years and the final amount remained the same?
3. What if the growth were compounded monthly instead of annually?
4. How long would it take for the investment to double with this growth rate?
5. How does the concept of continuous compounding affect the calculation of the growth rate?

**Tip:** When working with exponents and roots, use a calculator or software to ensure precision, especially with non-integer values.

Learn how to calculate the annual growth rate (y) when an investment grows from $8,000 to $10,000 in 3 years with annual compounding. This problem covers compound interest and exponential growth concepts, using the compound interest formula to determine the growth rate. Suitable for students in Grades 10-12.

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