Question

Mr. Bert deposited $5,000 into an investment account 20 years ago with an annual interest rate of 3.75%.

a) Find the exponential function in the form f(x)=a⋅bx
to represent the total value of Mr. Bert’s investment account over time.

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 the answers to two decimal places.

Answer 1

To solve the problems, we'll follow these steps:

### a) Finding the Exponential Function

The formula for compound interest can be represented as:

\[
A(t) = P(1 + r)^t
\]

where:
- \( A(t) \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal).
- \( t \) is the time in years.

For Mr. Bert's investment:
- \( P = 5000 \)
- \( r = 3.75\% = 0.0375 \)
- \( t \) is the time in years.

We want to express the function in the form \( f(x) = a \cdot b^x \).

In this case:
- The initial amount \( a = 5000 \).
- The base \( b = (1 + r) = (1 + 0.0375) = 1.0375 \).

Thus, the exponential function representing the total value of Mr. Bert's investment account over time is:

\[
f(t) = 5000 \cdot (1.0375)^t
\]

### b) Total Value of Investment When Mr. Bert Retires

Mr. Bert will retire in 15 years. Thus, the total time since the initial deposit will be:

\[
t = 20 + 15 = 35 \text{ years}
\]

Now we can calculate the total value at retirement:

\[
A(35) = 5000 \cdot (1.0375)^{35}
\]

Calculating \( (1.0375)^{35} \):

\[
(1.0375)^{35} \approx 3.455 (using a calculator)
\]

Now, substituting this value back into the function:

\[
A(35) = 5000 \cdot 3.455 \approx 17275
\]

So, the total value of the investment when Mr. Bert retires is approximately **$17,275.00**.

### Percent Increase

To find the percentage increase:

1. Calculate the increase:
\[
\text{Increase} = \text{Final Amount} - \text{Initial Amount} = 17275 - 5000 = 12275
\]

2. Calculate the percent increase:
\[
\text{Percent Increase} = \left( \frac{\text{Increase}}{\text{Initial Amount}} \times 100 \right) = \left( \frac{12275}{5000} \times 100 \right) \approx 245.5\%
\]

In conclusion:
- The total value of Mr. Bert’s investment when he retires is **$17,275.00**.
- The percent increase compared to the initial amount of the investment is approximately **245.50%**.