Find the interest rate for aa ​$44000 deposit accumulating to ​$49734973​, compounded annually for 77 years.
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Part 1
The interest rate is enter your response here​%.
​(Do not round until the final answer. Then round to two decimal places as​ needed.)

To solve for the annual interest rate $ r $ for a principal $ P = 44,000 $ to grow to an accumulated amount $ A = 49,734,973 $ over $ t = 77 $ years, compounded annually, we can use the compound interest formula:

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

### Step 1: Substitute the known values into the formula
\[
49,734,973 = 44,000(1 + r)^{77}
\]

### Step 2: Isolate $(1 + r)^{77}$
Divide both sides by 44,000:

\[
\frac{49,734,973}{44,000} = (1 + r)^{77}
\]

\[
1,130.34 = (1 + r)^{77}
\]

### Step 3: Take the 77th root of both sides to solve for $1 + r$
\[
1 + r = (1130.34)^{\frac{1}{77}}
\]

### Step 4: Calculate the result
Using a calculator or computational tool, find the 77th root of 1130.34:

\[
1 + r \approx 1.104
\]

### Step 5: Solve for $r$ by subtracting 1
\[
r \approx 1.104 - 1 = 0.104 \quad \text{or} \quad 10.4\%
\]

### Final Answer
The interest rate is approximately **10.4%**.

Would you like more detailed steps or have any other questions?

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Here are 5 additional questions to explore this topic further:
1. What happens to the accumulated amount if the interest rate is compounded monthly instead?
2. How would the interest rate change if the time period was only 50 years?
3. How does compound interest differ from simple interest in terms of growth over long periods?
4. What if the principal amount was doubled; how would that affect the accumulated total?
5. How sensitive is the accumulated amount to changes in the interest rate?

**Tip:** Always use the appropriate root (such as $ \frac{1}{t} $) when solving for the rate in compounded interest problems.

Find the interest rate for a $44,000 deposit accumulating to $49,734,973, compounded annually for 77 years. The interest rate is __%. (Do not round until the final answer. Then round to two decimal places as needed.)

Learn how to find the annual interest rate needed for a $44,000 deposit to grow to $49,734,973 over 77 years with annual compounding. This problem applies the compound interest formula and involves root extraction, suitable for high school students.

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