Sure, let's solve the problem of finding the final amount owed after 7 years when $2000 is loaned at an interest rate of 11.5%, compounded semiannually.
To solve this problem, we will use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount owed after the given time.
- [tex]\( P \)[/tex] is the principal amount (initial amount loaned).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is loaned for.
Given:
- [tex]\( P = 2000 \)[/tex] dollars
- [tex]\( r = 11.5\% = \frac{11.5}{100} = 0.115 \)[/tex] (in decimal form)
- [tex]\( n = 2 \)[/tex] (since the interest is compounded semiannually, i.e., twice per year)
- [tex]\( t = 7 \)[/tex] years
Substitute these values into the compound interest formula:
[tex]\[ A = 2000 \left(1 + \frac{0.115}{2}\right)^{2 \times 7} \][/tex]
First, calculate the interest per compounding period:
[tex]\[ \frac{0.115}{2} = 0.0575 \][/tex]
Next, compute the exponent:
[tex]\[ 2 \times 7 = 14 \][/tex]
Now, substitute these intermediate results back into the formula:
[tex]\[ A = 2000 \left(1 + 0.0575\right)^{14} \][/tex]
[tex]\[ A = 2000 \left(1.0575\right)^{14} \][/tex]
Determine the value of [tex]\( (1.0575)^{14} \)[/tex]:
[tex]\[ (1.0575)^{14} \approx 2.187385 \][/tex]
Finally, calculate the amount owed:
[tex]\[ A = 2000 \times 2.187385 \approx 4374.77 \][/tex]
So, the amount owed after 7 years, rounded to the nearest cent, is:
[tex]\[ \boxed{4374.77} \][/tex]
