To solve this problem, we can follow these steps:
Part a: Find the Exponential Function
The formula for the future value of an investment compounded annually is given by:
\[ f(t) = P(1 + r)^t \]
where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P = 5000 \),
- \( r = 3.75% = \frac{3.75}{100} = 0.0375 \),
- We denote \( t \) as the number of years since the initial deposit.
Thus, the function can be written as:
\[ f(t) = 5000(1 + 0.0375)^t \]
To express it in the form \( f(x) = a \cdot b^x \):
Here, \( a = 5000 \) and \( b = 1 + 0.0375 = 1.0375 \).
So the exponential function is:
\[ f(t) = 5000(1.0375)^t \]
Part b: Calculate the Total Value When Mr. Bert Retires
Mr. Bert will retire in 15 years; since the investment account was made 20 years ago, we find the total number of years from the initial deposit until retirement:
\[ \text{Total time until retirement} = 20 + 15 = 35 \text{ years} \]
Now, we can substitute \( t = 35 \) into the exponential function:
\[ f(35) = 5000(1.0375)^{35} \]
Now let's calculate \( (1.0375)^{35} \):
\[ (1.0375)^{35} \approx 3.578217 \]
Now substitute back into the function:
\[ f(35) = 5000 \cdot 3.578217 \approx 17891.09 \]
So, the total value of the investment when Mr. Bert retires is approximately:
\[ \text{Total Value} \approx 17891.09 \]
Percent Increase Calculation
To find the percent increase compared to the initial amount, we use the formula for percentage increase:
\[ \text{Percent Increase} = \frac{\text{Final Amount} - \text{Initial Amount}}{\text{Initial Amount}} \times 100 \]
Substituting the values:
\[ \text{Percent Increase} = \frac{17891.09 - 5000}{5000} \times 100 = \frac{12891.09}{5000} \times 100 \approx 257.82% \]
Final Answers:
- a) The exponential function is \( f(t) = 5000(1.0375)^t \).
- b) Total value at retirement is approximately $17,891.09 and the percent increase is 257.82%.