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Part 1
​(Related to Checkpoint​ 5.6) ​ (Solving for i​)  At what annual interest​ rate, compounded​ annually, would ​$500 have to be invested for it to grow to ​$1 comma 972.25 in 13 ​years?
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Part 1
The annual interest​ rate, compounded​ annually, at which ​$500 must be invested for it to grow to ​$1 comma 972.25 in 13 years is 
  
enter your response here​%.  ​(Round to two decimal​ places.)

We can solve this problem using the formula for compound interest:

\[
A = P(1 + r)^t
\]

Where:
- $ A $ is the final amount ($1972.25),
- $ P $ is the initial principal ($500),
- $ r $ is the annual interest rate (which we need to find),
- $ t $ is the time in years (13 years).

### Step-by-step solution:
1. Start with the compound interest formula:
   \[
   A = P(1 + r)^t
   \]
   Substituting the given values:
   \[
   1972.25 = 500(1 + r)^{13}
   \]

2. Divide both sides by 500:
   \[
   \frac{1972.25}{500} = (1 + r)^{13}
   \]
   Simplifying:
   \[
   3.9445 = (1 + r)^{13}
   \]

3. Take the 13th root of both sides to isolate $ r $:
   \[
   (3.9445)^{\frac{1}{13}} = 1 + r
   \]

4. Calculate the 13th root:
   \[
   3.9445^{\frac{1}{13}} \approx 1.115
   \]

5. Subtract 1 from both sides to solve for $ r $:
   \[
   r \approx 1.115 - 1 = 0.115
   \]

6. Convert the decimal to a percentage:
   \[
   r \approx 11.5\%
   \]

Thus, the annual interest rate is approximately **11.5%**.

Would you like more details on any step?

### Related Questions:
1. How does the compound interest formula differ from simple interest?
2. What happens to the interest rate if the compounding frequency changes?
3. How can we adjust the formula for monthly or quarterly compounding?
4. Can the formula be used to find the investment duration if the interest rate is known?
5. How can this be solved using logarithms instead of roots?

### Tip:
When dealing with compound interest, knowing how to manipulate exponents and roots is key to isolating variables like the interest rate or time.

At what annual interest rate, compounded annually, would $500 have to be invested for it to grow to $1,972.25 in 13 years? Round to two decimal places.

Learn how to calculate the annual interest rate needed for $500 to grow to $1,972.25 in 13 years using the compound interest formula. This problem covers key algebraic concepts, including exponential functions and solving for interest rates. Suitable for students in Grades 9-12.

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