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.