To solve the problem, we will start with part (a) to find the exponential function representing Mr. Bert's investment.
Part (a)
The formula for compound interest is given by:
\[ A(t) = P(1 + r)^t \]
Where:
- \( A(t) \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for in years.
In this case:
- \( P = 5000 \)
- \( r = 3.75% = 0.0375 \)
Thus, the function can be written as:
\[ A(t) = 5000(1 + 0.0375)^t \]
Simplifying that:
\[ A(t) = 5000(1.0375)^t \]
Now, in order to convert this into the form \( f(x) = a \cdot b^x \), we can use:
- \( a = 5000 \)
- \( b = 1.0375 \)
So the exponential function is:
\[ f(t) = 5000 \cdot (1.0375)^t \]
Part (b)
Now, we want to calculate the total value of the investment when Mr. Bert retires in 15 years after the 20 years already passed, which makes it a total of \( 20 + 15 = 35 \) years.
Let's calculate the total value of the investment after 35 years using the formula:
\[ A(35) = 5000(1.0375)^{35} \]
Calculating \( (1.0375)^{35} \):
\[ (1.0375)^{35} \approx 3.545 - \text{ (using a calculator)} \]
Now substituting into the equation:
\[ A(35) \approx 5000(3.545) \approx 17725 \]
So, the total value of Mr. Bert's investment when he retires is approximately $17,725.
Next, to find the percent increase compared to the initial amount:
Initial amount = $5,000
Final amount = $17,725
Percentage increase can be calculated as:
\[ \text{Percent Increase} = \left( \frac{\text{Final Amount} - \text{Initial Amount}}{\text{Initial Amount}} \right) \times 100 \]
Calculating:
\[ \text{Percent Increase} = \left( \frac{17725 - 5000}{5000} \right) \times 100 \]
\[ = \left( \frac{12725}{5000} \right) \times 100 \]
\[ \approx 254.50% \]
Final Answers:
a) The exponential function is:
\[ f(t) = 5000 \cdot (1.0375)^t \]
b) The total value of the investment when Mr. Bert retires is approximately $17,725.00 and the percent increase compared to the initial amount is approximately 254.50%.
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