To solve for $ r $ in the equation:

\[
1201 = 370 \times \left(1 + \frac{r}{100}\right)^{19}
\]

### Steps:

1. **Divide both sides by 370** to simplify the equation:
   \[
   \frac{1201}{370} = \left(1 + \frac{r}{100}\right)^{19}
   \]
   \[
   3.2473 \approx \left(1 + \frac{r}{100}\right)^{19}
   \]

2. **Take the 19th root** of both sides to remove the exponent:
   \[
   \sqrt[19]{3.2473} = 1 + \frac{r}{100}
   \]

3. **Subtract 1** from both sides to isolate $\frac{r}{100}$:
   \[
   \sqrt[19]{3.2473} - 1 = \frac{r}{100}
   \]

4. **Multiply both sides by 100** to solve for $ r $:
   \[
   r = 100 \times \left(\sqrt[19]{3.2473} - 1\right)
   \]

### Calculation:
Now, compute the value:

\[
\sqrt[19]{3.2473} \approx 1.0619
\]
\[
r \approx 100 \times (1.0619 - 1) = 100 \times 0.0619 = 6.19\%
\]

So, the value of $ r $ is approximately **6.19%**.

Would you like any further explanation, or do you have any questions?

### Relative Questions:
1. How does the length of the investment period affect the interest rate required to reach a certain amount?
2. Why is it necessary to take the 19th root in this equation?
3. How would the interest rate change if the final amount were different, say $1,500?
4. What are the practical applications of solving for the interest rate in financial planning?
5. How does this calculation differ when considering simple interest instead of compound interest?

### Tip:
When solving for the interest rate in compound interest problems, taking the root corresponding to the time period is essential to accurately determine the rate over that period.

Learn how to calculate the interest rate using the compound interest formula with a step-by-step guide. This problem involves finding the interest rate in a compound interest scenario based on given principal, final amount, and time period. Suitable for Grades 10-12.

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