To analyze the function \( f(t) = 4000(1.0056)^{12t} \):
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Initial Amount of Money Invested: The initial amount of money invested is represented by the coefficient of the function. In this case, it is: \[ \text{Initial amount} = 4000 , \text{dollars} \]
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Exponential Growth Rate: The term \( (1.0056)^{12t} \) indicates compound interest. The base of this exponent, which is \( 1.0056 \), can be converted to a percentage increase per time period (in this case, per month).
To find the annual growth rate, we can calculate: \[ (1.0056)^{12} \]
To simplify, let's compute this: \[ (1.0056)^{12} \approx 1.0673 \quad \text{(using a calculator)} \]
The annual growth rate, as a percentage, is then: \[ (1.0673 - 1) \times 100 \approx 6.73% \]
So, to summarize:
- The initial amount of money invested is $4,000.
- The function represents an exponential model where the amount of money in the account increases by approximately 6.73% each year.