To determine the amount owed after four years on a loan of [tex]$3900 at an interest rate of 5% per year, compounded continuously, we will use the formula for continuous compounding of interest:
\[ A = P \cdot e^{rt} \]
where:
- \( A \) is the amount owed after time \( t \),
- \( P \) is the principal, or initial amount of the loan,
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828,
- \( r \) is the annual interest rate (expressed as a decimal),
- \( t \) is the time the money is invested or borrowed for, in years.
Given:
- The principal \( P \) is $[/tex]3900,
- The annual interest rate [tex]\( r \)[/tex] is 5%, or 0.05 as a decimal,
- The time [tex]\( t \)[/tex] is 4 years.
Let's plug these values into the formula:
[tex]\[ A = 3900 \cdot e^{0.05 \cdot 4} \][/tex]
First, calculate the exponent:
[tex]\[ 0.05 \cdot 4 = 0.20 \][/tex]
Now, we need to calculate [tex]\( e^{0.20} \)[/tex]. Using the value:
[tex]\[ e^{0.20} \approx 1.22140275816 \][/tex]
Then, multiply this value by the principal [tex]\( 3900 \)[/tex]:
[tex]\[ A = 3900 \cdot 1.22140275816 \approx 4763.470756824662 \][/tex]
Next, we round the amount to the nearest cent. The result is:
[tex]\[ A \approx 4763.47 \][/tex]
Therefore, after four years, the amount owed on a loan of [tex]$3900 with an annual interest rate of 5%, compounded continuously, would be approximately $[/tex]4763.47.
