Answer :
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.
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.