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