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### Exponential and Logarithmic Functions

Finding the final amount in a word problem on compound interest:

Suppose that $2000 is loaned at a rate of 11.5%, compounded semiannually. Assuming that no payments are made, find the amount owed after 7 years.

Do not round any intermediate computations, and round your answer to the nearest cent.


Sagot :

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]