1
Question content area top

Part 1

​(Related **to Checkpoint**​ 5.6)**** ​ (Solving **for** 

***i***​)**** 

 At what annual interest​ rate, compounded​ annually, would 

​$490490 

have to be invested for it to grow to 

​$1 comma 935.151,935.15 

in 

1111 

​years?

Question content area bottom

Part 1

The annual interest​ rate, compounded​ annually, at which 

​$490490 

must be invested for it to grow to 

​$1 comma 935.151,935.15 

in 

1111 

years is 

enter your response here​%. 

 ​(Round to two decimal​ places.)

We are given an investment of \$490,490 and a final amount of \$1,935,151 after 11 years. The interest rate is compounded annually, and we need to determine the annual interest rate, $ r $, that achieves this growth.

We can use the compound interest formula:

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

Where:
- $ A $ is the final amount (\$1,935,151),
- $ P $ is the initial principal (\$490,490),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years (11 years).

### Step 1: Plug in the given values.

\[
1,935,151 = 490,490(1 + r)^{11}
\]

### Step 2: Solve for $ r $.

1. Divide both sides by 490,490:

\[
\frac{1,935,151}{490,490} = (1 + r)^{11}
\]

\[
3.945 = (1 + r)^{11}
\]

2. Take the 11th root of both sides to eliminate the exponent:

\[
(1 + r) = \left( 3.945 \right)^{\frac{1}{11}}
\]

3. Calculate $ \left( 3.945 \right)^{\frac{1}{11}} $:

\[
1 + r = 1.136
\]

4. Subtract 1 from both sides:

\[
r = 0.136
\]

### Step 3: Convert to a percentage.

\[
r = 0.136 \times 100 = 13.6\%
\]

### Final Answer:
The annual interest rate is approximately **13.60%**.

Would you like more details or have any questions? Here are 5 related questions to explore:

1. How does the formula change if the interest is compounded monthly instead of annually?
2. What would happen to the final amount if the investment period is shortened to 8 years?
3. How do we calculate the future value if the interest rate is variable over time?
4. What if the initial investment were \$1,000,000 instead—what interest rate would be required?
5. How does continuous compounding affect the required interest rate for the same final amount?

**Tip:** When using the compound interest formula, always ensure that the time period and interest compounding frequency align to avoid errors in calculation.

At what annual interest rate, compounded annually, would $490,490 have to be invested for it to grow to $1,935,151 in 11 years? (Round to two decimal places.)

Learn how to calculate the annual interest rate required for a $490,490 investment to grow to $1,935,151 in 11 years using the compound interest formula. This problem covers key concepts in finance, including exponential growth and interest rate calculations. Suitable for Grades 10-12.

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