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The problem asks to find the amount of time required for a $2000 investment to double under different compounding methods at two given interest rates: $ r = 4.75\% $ (Exercise 19) and $ r = 8.25\% $ (Exercise 20). The problem specifies four compounding methods: 

- (a) Annually,
- (b) Monthly,
- (c) Quarterly, 
- (d) Continuously.

To solve, we'll use the appropriate formulas for each compounding method:

### For periodic compounding (a, b, c):
The general formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $A$ is the final amount,
- $P$ is the initial principal (here $2000),
- $r$ is the annual interest rate (as a decimal),
- $n$ is the number of times the interest is compounded per year,
- $t$ is the time in years.

To find the time required to double the investment, set $ A = 2P $ and solve for $ t $:
\[
2 = \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
t = \frac{\log(2)}{n \log\left(1 + \frac{r}{n}\right)}
\]

### For continuous compounding (d):
The formula for continuous compounding is:
\[
A = P e^{rt}
\]
Setting $A = 2P$ gives:
\[
2 = e^{rt}
\]
\[
t = \frac{\log(2)}{r}
\]

Let me calculate the values for both exercises (19: $r = 4.75\%$ and 20: $r = 8.25\%$) under all four compounding methods.Here are the results for the time required to double a $2000 investment under different compounding methods for both interest rates:

### **Exercise 19 (r = 4.75%):**
- **Annually:** 14.94 years
- **Monthly:** 14.62 years
- **Quarterly:** 14.68 years
- **Continuously:** 14.59 years

### **Exercise 20 (r = 8.25%):**
- **Annually:** 8.74 years
- **Monthly:** 8.43 years
- **Quarterly:** 8.49 years
- **Continuously:** 8.40 years

Let me know if you'd like further details or clarifications.

Here are some related questions you might consider:
1. How does increasing the interest rate affect the time to double an investment?
2. Why does continuous compounding result in a slightly shorter time than periodic compounding?
3. What would happen to the doubling time if the investment were compounded daily?
4. How does the time to double change if the interest rate is compounded semi-annually?
5. How can you calculate the future value of an investment if it is compounded weekly?

**Tip:** When working with different compounding intervals, the more frequently the interest is compounded, the faster the investment grows.

In Exercises 19 and 20, find the amount of time required for a $2000 investment to double if the annual interest rate r is compounded (a) annually, (b) monthly, (c) quarterly, and (d) continuously.

Explore how long it takes for a $2000 investment to double with interest rates of 4.75% and 8.25%, under annual, monthly, quarterly, and continuous compounding. Learn through detailed formulas and calculations for exponential growth and compounding interest.

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