To find the actual interest rate and the future values for the given functions, let's break down the two functions you provided.
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First Function: \[ f(x) = 300(1.32)^{2t} \]
The term \( (1.32)^{2t} \) suggests that the investment is compounded, and with a base of \( 1.32 \), the growth factor is \( 1.32 \). To find the actual interest rate, we can use the formula for continuous compounding or find the effective annual rate. Here, \( 1.32 - 1 = 0.32 \) or \( 32% \) is the effective interest rate per compounding period.
Therefore, the actual interest rate is effectively expressed in terms of the growth factor:
- The annual interest rate in this case would be \( 32% \).
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Second Function: \[ f(x) = 300(1.32)^{t/2} \]
In this case, with \( (1.32)^{t/2} \), it implies that the growth is based on half the time variable, effectively meaning it is compounded every 2 years or with less frequent compounding period.
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Since we have the same base of \( 1.32 \), the effective growth factor for a full year would typically be calculated similarly, leading to similar results, but with less frequency of compounding.
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The actual rate can again be projected as:
- The effective interest rate here can also be found as \( 32% \).
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Future Value Calculation:
- For both functions, we will calculate the future value after 10 years.
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Future Value for \( f(x) = 300(1.32)^{2t} \) for \( t = 10 \): \[ f(10) = 300(1.32)^{20} \] \[ = 300 \times (1.32)^{20} \] \[ = 300 \times 37.949 , (\text{using a calculator for } (1.32)^{20} \approx 37.949) \] \[ = 11,384.70 \]
While it seems mismatched for these listed values, ensure it's verified against options.
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Future Value for \( f(x) = 300(1.32)^{t/2} \) for \( t = 10 \): \[ f(10) = 300(1.32)^{10} \] \[ = 300 \times (1.32)^{5} , (\text{since } t/2 = 5) \] \[ \approx 300 \times 4.36 , (\text{using a calculator for } (1.32)^{5} \approx 4.36) \] \[ = 1,308.00 \]
Thus, the answers will be taken from future value possibilities.
Final Answers:
- Actual Interest Rate: 32%
- Future Value (for 10 years under both models): Confirm \( 1.32^{20} \) = Approx \( 11,384.70 \) and for \( 10 \) years old model into those pots.
Depending on additional contexts or incorrect inputs could imply double-checking \( t/2 \) vs \( 2t \) function accumulations and answering how they're arranged.
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