To find the average amount earned per year between the 20th year and the 25th year for a continuous compounding interest investment, we will use the formula for continuous compounding given by:
\[ A(t) = Pe^{rt} \]
where:
- \( A(t) \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
In this case:
- \( P = 5000 \)
- \( r = 0.04 \)
First, we will calculate the amount of money at the end of 20 years (\( A(20) \)) and at the end of 25 years (\( A(25) \)).
- Calculate \( A(20) \):
\[ A(20) = 5000 e^{0.04 \times 20} \]
Calculating \( 0.04 \times 20 = 0.8 \), we have:
\[ A(20) = 5000 e^{0.8} \approx 5000 \times 2.22554 \approx 11127.70 \]
- Calculate \( A(25) \):
\[ A(25) = 5000 e^{0.04 \times 25} \]
Calculating \( 0.04 \times 25 = 1.0 \), we have:
\[ A(25) = 5000 e^{1.0} \approx 5000 \times 2.71828 \approx 13591.40 \]
Now that we have \( A(20) \) and \( A(25) \):
- \( A(20) \approx 11127.70 \)
- \( A(25) \approx 13591.40 \)
Next, we will find the total amount earned between the 20th and 25th year:
\[ \text{Amount earned between 20 and 25 years} = A(25) - A(20) \approx 13591.40 - 11127.70 = 2463.70 \]
Finally, we will find the average amount earned per year from year 20 to year 25:
Since this period spans 5 years (from year 20 to year 25), we compute:
\[ \text{Average amount earned per year} = \frac{\text{Total amount earned}}{\text{Number of years}} = \frac{2463.70}{5} \approx 492.74 \]
Thus, the average amount earned per year between the 20th year and the 25th year is:
\[ \boxed{492.74} \]