To find the average amount earned per year between the 5th year and the 10th year for an investment of $5000 compounded continuously at 6%, we first need to use the formula for continuous compounding:
\[ A(t) = Pe^{rt} \]
Where:
- \( A(t) \) = amount of money accumulated after time \( t \) (in years)
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time in years
- \( e \) = Euler's number (approximately 2.71828)
Given:
- \( P = 5000 \)
- \( r = 0.06 \)
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Calculate the amount after 5 years: \[ A(5) = 5000 e^{0.06 \cdot 5} \] \[ A(5) = 5000 e^{0.3} \] Using a calculator to find \( e^{0.3} \): \[ e^{0.3} \approx 1.34986 \] So, \[ A(5) = 5000 \cdot 1.34986 \approx 6749.30 \]
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Calculate the amount after 10 years: \[ A(10) = 5000 e^{0.06 \cdot 10} \] \[ A(10) = 5000 e^{0.6} \] Using a calculator to find \( e^{0.6} \): \[ e^{0.6} \approx 1.82212 \] So, \[ A(10) = 5000 \cdot 1.82212 \approx 9110.61 \]
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Determine the total amount earned between the 5th year and the 10th year: The total earned between these two years is: \[ \text{Total earned} = A(10) - A(5) \] \[ \text{Total earned} = 9110.61 - 6749.30 \approx 2361.31 \]
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Calculate the average amount earned per year between the 5th year and the 10th year: The average amount earned per year over the 5 years (from year 5 to year 10) is: \[ \text{Average per year} = \frac{\text{Total earned}}{\text{Number of years}} = \frac{2361.31}{5} \approx 472.26 \]
Thus, the average amount earned per year between the 5th year and the 10th year is approximately $472.26.