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?

---

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.