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If [tex][tex]$\$[/tex]7000$[/tex] is invested at [tex]4\%[/tex] compounded continuously, what is the amount after 8 years?

Which of the following is the appropriate formula with the correct values substituted? Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A. [tex]A = Pe^{rt}[/tex], with [tex]P =[/tex] [tex]\square[/tex], [tex]r =[/tex] [tex]\square[/tex], and [tex]t =[/tex] [tex]\square[/tex]

B. [tex]A = P(1+i)^n[/tex], with [tex]P =[/tex] [tex]\square[/tex], [tex]i =[/tex] [tex]\square[/tex], and [tex]n =[/tex] [tex]\square[/tex]

C. [tex]A = P(1+rt)[/tex], with [tex]P =[/tex] [tex]\square[/tex], [tex]r =[/tex] [tex]\square[/tex], and [tex]t =[/tex] [tex]\square[/tex]

The amount after 8 years will be [tex]\$ \square[/tex].

(Round to the nearest cent.)



Answer :

To solve this problem, you need to calculate the amount of money accumulated after a certain time when the interest is compounded continuously. The appropriate formula for continuous compounding is:

[tex]\[ A = P e^{rt} \][/tex]

where:
- [tex]\( A \)[/tex] is the amount of money accumulated after time [tex]\( t \)[/tex],
- [tex]\( P \)[/tex] is the initial principal balance (initial investment),
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal),
- [tex]\( t \)[/tex] is the time in years,
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).

Given the values:
- [tex]\( P = 7000 \)[/tex] (the initial investment),
- [tex]\( r = 0.04 \)[/tex] (the interest rate, 4% expressed as a decimal),
- [tex]\( t = 8 \)[/tex] years,

we substitute these values into the formula:

[tex]\[ A = 7000 e^{0.04 \times 8} \][/tex]

Using this formula, the amount [tex]\( A \)[/tex] after 8 years is found to be:

[tex]\[ A \approx 9639.89 \][/tex]

So, for the multiple-choice question:

Choose the correct formula and values:
A. [tex]\( A = Pe^{rt} \)[/tex], with [tex]\( P = 7000 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 8 \)[/tex]

Therefore, the amount after 8 years will be \$ 9639.89.