To find the amount owed on a loan of [tex]$1800 after three years with an interest rate of 8.5% per year, compounded continuously, we will use the formula for continuous compounding. The formula is:
\[ A = P \cdot e^{rt} \]
where:
- \( A \) is the amount owed after time \( t \),
- \( P \) is the principal amount (initial loan amount),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time the money is borrowed for (in years),
- \( e \) is the base of the natural logarithm (approximately 2.71828).
Let's apply the values given in the problem:
1. \( P = 1800 \) (the principal amount),
2. \( r = 8.5\% = 0.085 \) (the annual interest rate in decimal form),
3. \( t = 3 \) (the time in years).
Substitute these values into the formula:
\[ A = 1800 \cdot e^{(0.085 \cdot 3)} \]
First, calculate the exponent:
\[ 0.085 \cdot 3 = 0.255 \]
Next, find \( e^{0.255} \). Using \( e \approx 2.71828 \):
\[ e^{0.255} \approx 1.29033 \]
Now, multiply this result by the principal amount to find the amount owed:
\[ A = 1800 \cdot 1.29033 \approx 2322.83 \]
So, the amount owed after three years, rounded to the nearest cent, is $[/tex]2322.83.
