To solve this problem, we'll use two different compound interest formulas based on the type of compounding (monthly and continuously). We are given:

• Principal (P) = $26
• Annual interest rate (r) = 6% or 0.06
• Time (t) = 2010 - 1885 = 125 years

Part 1: Monthly Compounding

The formula for monthly compounding is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ Where:

• $A$ is the amount of money accumulated after interest.
• $P$ is the principal amount ($26).
• $r$ is the annual interest rate (0.06).
• $n$ is the number of times interest is compounded per year (12 for monthly).
• $t$ is the time the money is invested for (125 years).

Substituting the given values:

$A = 26 \left(1 + \frac{0.06}{12}\right)^{12 \times 125}$ Now, let’s calculate it:

$A = 26 \left(1 + 0.005\right)^{1500}$ $A = 26 \times (1.005)^{1500}$

Using a calculator, we get:

$A \approx 46,139$

So, the investment would be worth $46,139 if compounded monthly.

Part 2: Continuous Compounding

The formula for continuous compounding is: $A = P \cdot e^{rt}$ Where:

• $A$ is the amount of money accumulated after interest.
• $P$ is the principal amount ($26).
• $r$ is the annual interest rate (0.06).
• $t$ is the time the money is invested for (125 years).
• $e$ is the base of the natural logarithm (approximately 2.71828).

Substituting the values:

$A = 26 \cdot e^{0.06 \times 125}$ $A = 26 \cdot e^{7.5}$

Using a calculator:

$A \approx 26 \times 1,823.293$ $A \approx 47,466$

So, the investment would be worth approximately $47,466 if compounded continuously.

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Final Answers:

• a. Monthly compounding: The investment would be worth $46,139.
• b. Continuous compounding: The investment would be worth $47,466.

Would you like further details or clarifications? Here are some related questions you might find interesting:

1. What would the investment be worth if the interest rate was 5% instead of 6%?
2. How would the compounding frequency affect the total amount if the investment was for only 50 years instead of 125?
3. What is the effect of increasing the compounding frequency (quarterly, daily, etc.) on the total accumulated value?
4. How would this investment look if the initial amount was $1000 instead of $26?
5. What are some real-world examples of continuously compounded interest?

Tip: When calculating continuous compounding, remember the key formula involves the natural exponential function $e$, which represents continuous growth.