Sure, let's solve this word problem step-by-step.
To find the final amount owed when interest is compounded continuously, we can use the formula for continuous compound interest, which is:
[tex]\[ A = P \cdot e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount owed at the end.
- [tex]\( P \)[/tex] is the principal amount, which is the initial amount borrowed.
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( t \)[/tex] is the time the money is borrowed for, in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given the information in the problem:
- The principal amount ([tex]\( P \)[/tex]) is $1100.
- The interest rate ([tex]\( r \)[/tex]) is 6% per year, which as a decimal, is 0.06.
- The time ([tex]\( t \)[/tex]) is 5 years.
Now we'll plug these values into the formula:
[tex]\[ A = 1100 \cdot e^{0.06 \cdot 5} \][/tex]
First, we calculate the exponent:
[tex]\[ 0.06 \cdot 5 = 0.3 \][/tex]
So the equation now looks like this:
[tex]\[ A = 1100 \cdot e^{0.3} \][/tex]
Next, we calculate [tex]\( e^{0.3} \)[/tex]:
[tex]\[ e^{0.3} \approx 1.34986 \][/tex]
Now, we'll multiply this by the principal amount:
[tex]\[ A = 1100 \cdot 1.34986 \approx 1484.846 \][/tex]
Finally, we round the result to the nearest cent:
[tex]\[ A \approx 1484.84 \][/tex]
Therefore, the amount owed at the end of five years is:
[tex]\[ \boxed{1484.84} \][/tex]
This is the amount Emily would owe after five years, assuming no payments are made until the end and the interest is compounded continuously.
