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

Question

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

Let's solve the problem step by step.

Part a: Finding the Exponential Function

To find the exponential function in the form \( f(x) = a \cdot b^x \):

1. Identify the initial deposit and growth factor:

   • Initial deposit \( a = 5000 \)
   • Annual interest rate \( r = 3.75% = 0.0375 \)

2. The base \( b \) for the growth is calculated by: \[ b = 1 + r = 1 + 0.0375 = 1.0375 \]

3. The time variable \( x \) represents the number of years after the initial deposit. Since Mr. Bert deposited the money 20 years ago, we want to express this generally, so we will set the function with \( x \) just representing how many years after the first deposit.

Thus, the exponential function representing the value of Mr. Bert's investment account over time is: \[ f(x) = 5000 \cdot (1.0375)^x \]

Part b: Calculating the Value When Mr. Bert Retires

To find the total value of the investment when Mr. Bert retires in 15 years, we have to evaluate the function at \( x = 35 \) (since 20 years have already passed, and he is retiring 15 years from now):

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

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

\[ (1.0375)^{35} \approx 3.44956 \quad \text{(using a calculator)} \]

Now, substituting this back into the function:

\[ f(35) \approx 5000 \cdot 3.44956 \approx 17247.80 \]

Final Amount at Retirement

Mr. Bert's total value in the investment account when he retires after 15 additional years will be approximately $17,247.80.

Calculating the Percent Increase

To find the percent increase compared to the original investment:

1. The original investment was $5,000. The final amount is approximately $17,247.80.

2. The increase in value is: \[ \text{Increase} = 17247.80 - 5000 = 12247.80 \]

3. The percent increase is calculated as: \[ \text{Percent Increase} = \left(\frac{\text{Increase}}{\text{Original Amount}}\right) \times 100 = \left(\frac{12247.80}{5000}\right) \times 100 \approx 244.96% \]

Summary of Results

• The exponential function is: \[ f(x) = 5000 \cdot (1.0375)^x \]
• The total value of Mr. Bert's investment when he retires will be approximately $17,247.80.
• The percent increase of this final amount compared to the initial amount will be approximately 244.96%.