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Determine the amount of money, to the nearest cent, in the account after 15 years using the given information:

A person places [tex]\$ 52700[/tex] in an investment account earning an annual rate of [tex]3.2\%[/tex], compounded continuously.

Use the formula [tex]V = P e^{rt}[/tex], where:
- [tex]V[/tex] is the value of the account in [tex]t[/tex] years,
- [tex]P[/tex] is the principal initially invested,
- [tex]e[/tex] is the base of the natural logarithm,
- [tex]r[/tex] is the rate of interest,
- [tex]t[/tex] is the number of years.



Answer :

GinnyAnswer
Certainly! Let's solve the problem step-by-step.

We are given the following information:
- Initial principal, [tex]\( P = \$52700 \)[/tex]
- Annual interest rate, [tex]\( r = 3.2\% \)[/tex] (expressed as a decimal [tex]\( r = 0.032 \)[/tex])
- Time, [tex]\( t = 15 \)[/tex] years

We need to find the value of the account after 15 years using the continuously compounded interest formula:

[tex]\[ V = P e^{r t} \][/tex]

where:
- [tex]\( V \)[/tex] is the value of the account after time [tex]\( t \)[/tex].
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for.

Following the steps:

1. Substitute the known values into the formula:
[tex]\[ V = 52700 \cdot e^{0.032 \cdot 15} \][/tex]

2. Calculate the exponent:
[tex]\[ 0.032 \cdot 15 = 0.48 \][/tex]

3. Raise [tex]\( e \)[/tex] to the power of 0.48:
[tex]\[ e^{0.48} \][/tex]

4. Multiply the result by the principal:
[tex]\[ V = 52700 \cdot e^{0.48} \][/tex]

Using the value given by the calculation for [tex]\( e^{0.48} \)[/tex]:

[tex]\[ e^{0.48} \approx 1.615540 \][/tex]

Hence,

[tex]\[ V = 52700 \cdot 1.615540 \approx 85167.12099556548 \][/tex]

5. Round to the nearest cent:
[tex]\[ V \approx 85167.12 \][/tex]

Therefore, after 15 years, the amount of money in the investment account, to the nearest cent, is \$85167.12.

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