Certainly! We need to determine the amount owed after five years on a loan of [tex]$1500, given that the interest rate is 2% per year and it is compounded continuously.
To solve this problem, we will use the formula for continuous compounding interest, which is:
\[ A = P \cdot e^{rt} \]
where:
- \( A \) is the amount owed.
- \( P \) is the principal amount (initial loan).
- \( r \) is the annual interest rate (expressed as a decimal).
- \( t \) is the time the money is invested or borrowed for, in years.
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
Given:
- \( P = 1500 \) (principal amount)
- \( r = 0.02 \) (2% annual interest rate)
- \( t = 5 \) (time in years)
Let's plug these values into the continuous compounding formula:
1. Substitute the values into the formula:
\[ A = 1500 \cdot e^{0.02 \times 5} \]
2. Compute the exponent:
\[ 0.02 \times 5 = 0.1 \]
3. Find the value of \( e^{0.1} \). (Using a scientific calculator or logarithmic tables, we find \( e^{0.1} \approx 1.10517 \))
4. Now, multiply the principal amount by this factor:
\[ A = 1500 \cdot 1.10517 \]
5. Perform the multiplication:
\[ A \approx 1657.7563771134717 \]
6. Finally, round the amount to the nearest cent:
\[ A \approx 1657.76 \]
Therefore, the amount owed after five years on a $[/tex]1500 loan at an interest rate of 2% per year, compounded continuously, is approximately $1657.76.
