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To solve the given problems regarding compound interest, let's go through them one by one.

### Formula for Compound Interest:
For **monthly** compounding, we use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ = future value
- $P$ = principal (initial investment)
- $r$ = annual interest rate (in decimal form)
- $n$ = number of times interest is compounded per year
- $t$ = time (in years)

For **continuous** compounding, we use the formula:

\[
A = P e^{rt}
\]

Where:
- $e$ is Euler's number (approximately 2.718)
- Other variables are the same.

Given values:
- $P = 12,000$
- $r = 0.04$ (4% annual interest rate)
- $t = 5$ years

---

### Part (a): Monthly Compounding
In this case, the interest is compounded monthly. So, $n = 12$ (12 months in a year). Using the compound interest formula:

\[
A = 12,000 \left(1 + \frac{0.04}{12}\right)^{12 \times 5}
\]

Let's calculate the future value.

### Part (b): Continuous Compounding
For continuous compounding, we use:

\[
A = 12,000 e^{0.04 \times 5}
\]

Let's calculate this as well.

### Part (c): Time to Double (Continuous Compounding)
To find how long it will take for the money to double, we set $A = 2P$, so the equation becomes:

\[
2P = P e^{rt}
\]

Dividing both sides by $P$:

\[
2 = e^{rt}
\]

Taking the natural logarithm (ln) of both sides:

\[
\ln(2) = rt
\]

Solving for $t$:

\[
t = \frac{\ln(2)}{r}
\]

Substitute $r = 0.04$ to find the time.

---

### Part (d): 4-Month Term (Continuous Compounding)
For a term of 4 months, we convert the time to years ($t = 4/12 = 1/3$ years). Using the continuous compounding formula:

\[
A = 12,000 e^{0.04 \times \frac{1}{3}}
\]

Now, I'll calculate all these values.Here are the results:

### (a) Monthly Compounding:
The future value of the investment after 5 years, compounded monthly, is approximately **$14,651.96**.

### (b) Continuous Compounding:
The future value of the investment after 5 years, compounded continuously, is approximately **$14,656.83**.

### (c) Time for Money to Double (Continuous Compounding):
It will take approximately **17.33 years** for the money to double with continuous compounding at 4% interest.

### (d) 4-Month Term (Continuous Compounding):
The future value after a 4-month term, compounded continuously, is approximately **$12,161.07**.

Let me know if you need any more details or have questions.

---

Here are 5 related questions to expand on this:
1. How would the future value change if the interest rate was 6% instead of 4%?
2. What is the effect of changing the compounding frequency (e.g., quarterly or daily) on future value?
3. How does the time to double change with different interest rates, like 3% or 5%?
4. Can we calculate the future value for other principal amounts, such as $15,000 or $20,000?
5. What is the impact of compounding for different terms like 10 years or 2 years?

**Tip:** Continuous compounding yields slightly higher values than monthly compounding due to the more frequent application of interest.

If $12,000 is invested for 5 years at 4%, find the future value at the end of 5 years if interest is compounded: Monthly, Continuously, How long for the money to double with continuous compounding, and for a 4-month term.

Solve for the future value of $12,000 invested for 5 years at 4% with monthly and continuous compounding. Also, calculate the time to double the investment and the value for a 4-month term using continuous compounding.

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