Puppose $$ \$ 20,000 $$ is invested at an annual rate of $$ 6 \% $$ for 10 years. Find the future value if interest is compounded as follows. Question $$ 19,5.1 .59-\mathrm{BE} $$ a. Annually b. Quarterly c. Monthly d. Daily ( 365 days) e. Continuously

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**Given:**  
$$ P = \$20{,}000 $$, annual rate $$ r = 6\% = 0.06 $$, and time $$ t = 10 $$ years.

The general compound interest formula is  
$$
A = P \left(1 + \frac{r}{m}\right)^{m t},
$$  
where $$ m $$ is the number of compounding periods per year.

For continuous compounding, the formula is  
$$
A = P e^{rt}.
$$

---

**a. Compounded Annually ($$ m = 1 $$)**

$$
A = 20000 \left(1 + \frac{0.06}{1}\right)^{1 \cdot 10} = 20000 \times (1.06)^{10}.
$$

Calculating $$ (1.06)^{10} \approx 1.79085 $$:

$$
A \approx 20000 \times 1.79085 \approx \$35{,}817.
$$

---

**b. Compounded Quarterly ($$ m = 4 $$)**

$$
A = 20000 \left(1 + \frac{0.06}{4}\right)^{4 \cdot 10} = 20000 \times (1.015)^{40}.
$$

Estimating $$ (1.015)^{40} \approx 1.81407 $$:

$$
A \approx 20000 \times 1.81407 \approx \$36{,}281.
$$

---

**c. Compounded Monthly ($$ m = 12 $$)**

$$
A = 20000 \left(1 + \frac{0.06}{12}\right)^{12 \cdot 10} = 20000 \times (1.005)^{120}.
$$

Estimating $$ (1.005)^{120} \approx 1.819 $$:

$$
A \approx 20000 \times 1.819 \approx \$36{,}380.
$$

---

**d. Compounded Daily ($$ m = 365 $$)**

$$
A = 20000 \left(1 + \frac{0.06}{365}\right)^{365 \cdot 10} = 20000 \times \left(1 + \frac{0.06}{365}\right)^{3650}.
$$

It is known that for large $$ m $$,  
$$
\left(1 + \frac{0.06}{365}\right)^{365} \approx e^{0.06},
$$  
thus, for 10 years,
$$
A \approx 20000 \times e^{0.06 \cdot 10} = 20000 \times e^{0.6}.
$$

Calculating $$ e^{0.6} \approx 1.82212 $$:

$$
A \approx 20000 \times 1.82212 \approx \$36{,}442.
$$

---

**e. Compounded Continuously**

$$
A = 20000 \, e^{0.06 \cdot 10} = 20000 \, e^{0.6}.
$$

Again, using $$ e^{0.6} \approx 1.82212 $$:

$$
A \approx 20000 \times 1.82212 \approx \$36{,}442.
$$

---

**Summary of Answers:**

- **Annually:** $$ \$35{,}817 $$
- **Quarterly:** $$ \$36{,}281 $$
- **Monthly:** $$ \$36{,}380 $$
- **Daily (365 days):** $$ \$36{,}442 $$
- **Continuously:** $$ \$36{,}442 $$
The future value of \$20,000 invested at 6% for 10 years is approximately: - Annually: \$35,817 - Quarterly: \$36,281 - Monthly: \$36,380 - Daily: \$36,442 - Continuously: \$36,442

Puppose is invested at an annual rate of for 10 years. Find the future value if interest is compounded as follows.
Question
a. Annually
b. Quarterly
c. Monthly
d. Daily ( 365 days)
e. Continuously

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Puppose is invested at an annual rate of for 10 years. Find the future value if interest is compounded as follows. Question a. Annually b. Quarterly c. Monthly d. Daily ( 365 days) e. Continuously