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Let's break down the problem into two parts and solve it step by step.

### Given:
- **Initial investment** = $5,000
- **Annual interest rate** = 3.75% (or 0.0375 in decimal)
- **Time** (initial deposit was made 20 years ago and Mr. Bert will retire in 15 years) = 35 years in total.

#### Part a) Find the exponential function in the form $ f(x) = a \cdot b^x $ to represent the total value of Mr. Bert's investment over time.

The exponential growth formula for compound interest is typically written as:
\[
A(t) = P \cdot (1 + r)^t
\]
Where:
- $ A(t) $ is the amount after time $ t $.
- $ P $ is the principal (initial amount).
- $ r $ is the interest rate per period.
- $ t $ is the time in years.

In this case:
- $ P = 5000 $
- $ r = 0.0375 $
- $ t $ is the number of years since the investment started.

We can rewrite this in the form $ f(x) = a \cdot b^x $, where:
- $ a = 5000 $
- $ b = 1 + r = 1 + 0.0375 = 1.0375 $
- $ x = t $ is the number of years.

Thus, the exponential function becomes:
\[
f(x) = 5000 \cdot (1.0375)^x
\]

This is the required function.

---

#### Part b) Mr. Bert will retire in 15 years. What will be the total value of his investment when he retires? What is the percent increase of this final amount compared to the initial amount of the investment? Round answers to two decimal places.

At retirement, Mr. Bert will have been investing for a total of 35 years (20 years already passed + 15 more years).

To find the total value at retirement:
\[
A(35) = 5000 \cdot (1.0375)^{35}
\]

Let me calculate this for you:
\[
A(35) = 5000 \cdot (1.0375)^{35} \approx 5000 \cdot 3.69754 = 18,487.72
\]

So, the total value of the investment when Mr. Bert retires will be approximately **$18,487.72**.

---

##### Percent Increase:

The percent increase is calculated by:
\[
\text{Percent Increase} = \frac{A(35) - P}{P} \times 100
\]

Substitute values:
\[
\text{Percent Increase} = \frac{18,487.72 - 5000}{5000} \times 100 = \frac{13,487.72}{5000} \times 100 \approx 269.75\%
\]

Thus, the percent increase is **269.75%**.

---

### Final Answers:
1. Exponential function: $ f(x) = 5000 \cdot (1.0375)^x $
2. Total value after 35 years: **$18,487.72**
3. Percent increase: **269.75%**

Do you have any questions or need further clarification? Here are some related questions for practice:

1. What would the value of the investment be after 30 years instead of 35?
2. If the interest rate were 4%, how would the exponential function change?
3. How much total interest would Mr. Bert have earned by retirement?
4. What would happen to the total value if Mr. Bert withdrew $1000 after 10 years?
5. How does increasing the initial deposit impact the growth over time?

**Tip:** Compound interest grows exponentially, so even small increases in the rate or time can significantly impact the final amount.

Mr. Bert deposited $5,000 into an investment account 20 years ago with an annual interest rate of 3.75%. Find the exponential function in the form f(x) = a·b^x to represent the total value of Mr. Bert's investment account over time. Mr. Bert will retire in 15 years. What will be the total value of his investment when he retires? What is the percent increase of this final amount compared to the initial amount of the investment?

Learn how to find the exponential function and calculate the future value of a $5,000 investment growing at 3.75% annually. See how to compute the final value after 35 years and the percent increase compared to the initial deposit.

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