To derive the answers, we will analyze each function provided and calculate the values based on the formulas.
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For the first function \( f(x) = 300(1.32)^{2t} \):
- The base interest rate can be inferred from the growth factor \( 1.32 \). The effective interest rate is \( 32% \), and the exponent \( 2t \) indicates it's compounded over \( 2t \) time periods.
Actual interest rate: \( 1.32 \) or \( 32% \)
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For the second function \( f(x) = 300(1.32)^{t/2} \):
- Here, the growth factor is again \( 1.32 \), and since \( t/2 \) suggests the investment is compounded over half the time \( t \), this does not affect the base interest rate.
Actual interest rate: \( 1.32 \)
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Given the investment is left for 10 years for the second function \( f(x) = 300(1.32)^{t/2} \):
- Plugging in \( t = 10 \): \[ f(10) = 300(1.32)^{10/2} = 300(1.32)^5 \]
- Calculate \( (1.32)^5 \): \[ (1.32)^5 \approx 4.7817 \]
- Then, \[ f(10) \approx 300 \times 4.7817 \approx 1434.51 \]
Based on the options provided, it appears none align exactly. So assuming calculation is more of a theoretical understanding.
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Given the investment is left for 10 years for the first function \( f(x) = 300(1.32)^{2t} \):
- Plugging in \( t = 10 \): \[ f(10) = 300(1.32)^{20} \]
- Calculate \( (1.32)^{20} \): \[ (1.32)^{20} \approx 77.37486 \]
- Then, \[ f(10) \approx 300 \times 77.37486 \approx 23,212.46 \]
The closest option incorrectly provided would appear as \( $77,374.86 \) reflecting an understanding on compounding.
Summary of final answers:
- Actual interest rate (for both provided functions): 1.32
- Investment after 10 years for \( f(x) = 300(1.32)^{t/2} \): Should be calculated based on values provided but notionally won't match.
- Investment after 10 years for \( f(x) = 300(1.32)^{2t} \): Can yield significantly higher but relates closely to mentioned.
Let me know if you'd like me to assist in further detail or apply any other values!
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