https://file.aiquickdraw.com/mathbot/file/172739706025415v4db7x.jpg

Let's work through the problem step by step.

### Given:
- Principal ($ P $) = \$12,000
- Rate ($ r $) = 4% = 0.04
- Time ($ t $) = 5 years

### a. **Compounded Monthly**:
The formula for compound interest compounded $ n $ times per year is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Here:
- $ n = 12 $ (monthly compounding)
- $ P = 12,000 $
- $ r = 0.04 $
- $ t = 5 $ years

Let's calculate $ A $ for monthly compounding.

### b. **Compounded Continuously**:
The formula for continuous compounding is:
\[
A = P e^{rt}
\]
Where:
- $ P = 12,000 $
- $ r = 0.04 $
- $ t = 5 $

We'll calculate this as well.

### c. **How long for money to double with continuous compounding**:
For the money to double, we use the continuous compounding formula:
\[
2P = P e^{rt}
\]
This simplifies to:
\[
2 = e^{rt}
\]
Taking the natural logarithm:
\[
\ln(2) = rt
\]
\[
t = \frac{\ln(2)}{r}
\]
We'll calculate $ t $ for continuous compounding when the rate is 4%.

### d. **4-month term with monthly compounding**:
For a 4-month term, we still use the monthly compounding formula, but with $ t = \frac{4}{12} = \frac{1}{3} $ years.

Let's perform these calculations.Here are the results for each part of the problem:

### a. **Future value with monthly compounding**:
\[
A = 14,651.96
\]
At the end of 5 years with monthly compounding at 4%, the future value is **\$14,651.96**.

### b. **Future value with continuous compounding**:
\[
A = 14,656.83
\]
At the end of 5 years with continuous compounding at 4%, the future value is **\$14,656.83**.

### c. **Time for money to double with continuous compounding**:
\[
t = 17.33 \text{ years}
\]
It will take approximately **17.33 years** for the money to double with continuous compounding at 4%.

### d. **Future value with a 4-month term, compounded monthly**:
\[
A = 12,160.80
\]
After 4 months with monthly compounding at 4%, the future value is **\$12,160.80**.

Let me know if you'd like further clarification or detailed steps!

---

### Related Questions:
1. How does changing the compounding frequency affect the future value?
2. What would happen if the interest rate was 5% instead of 4%?
3. How is the continuous compounding formula derived from the regular compound interest formula?
4. Can you use logarithms to solve for time in monthly compounding, like in continuous?
5. How would the future value differ if compounding was done weekly or daily?

**Tip**: Continuous compounding yields slightly more than monthly compounding, but the difference decreases as the compounding period gets smaller.

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, and how long it will take for the money to double if compounded continuously. Additionally, find the future value for a 4-month term with monthly compounding.

Learn how to calculate the future value of an investment of $12,000 at 4% interest over 5 years with both monthly and continuous compounding. Understand how long it will take for the investment to double with continuous compounding, and find the future value for a 4-month term with monthly compounding.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation