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How much would [tex]$\$[/tex]200[tex]$ invested at $[/tex]5\%[tex]$ interest compounded monthly be worth after 9 years? Round your answer to the nearest cent.

The formula to use is:
\[A(t) = P\left(1+\frac{r}{n}\right)^{nt}\]

A. $[/tex]\[tex]$313.37$[/tex]
B. [tex]$\$[/tex]207.63[tex]$
C. $[/tex]\[tex]$363.82$[/tex]
D. [tex]$\$[/tex]310.27$


Sagot :

To determine how much \[tex]$200 invested at 5% interest compounded monthly would be worth after 9 years, we use the formula for compound interest: \[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \(P\) is the principal amount (the initial investment), - \(r\) is the annual interest rate (as a decimal), - \(n\) is the number of times interest is compounded per year, - \(t\) is the number of years the money is invested, - \(A(t)\) is the amount of money accumulated after \(t\) years, including interest. Let's break down the values given: - \(P = \$[/tex]200\)
- [tex]\(r = 0.05\)[/tex] (5% annual interest rate)
- [tex]\(n = 12\)[/tex] (since interest is compounded monthly)
- [tex]\(t = 9\)[/tex] years

Now, substitute these values into the formula:

[tex]\[ A(9) = 200 \left(1 + \frac{0.05}{12}\right)^{12 \times 9} \][/tex]

First, calculate the monthly interest rate:

[tex]\[ \frac{0.05}{12} \approx 0.004167 \][/tex]

Next, compute the exponent:

[tex]\[ 12 \times 9 = 108 \][/tex]

Now calculate the expression inside the parentheses:

[tex]\[ 1 + 0.004167 \approx 1.004167 \][/tex]

Raise this to the power of 108:

[tex]\[ (1.004167)^{108} \approx 1.566298 \][/tex]

Finally, multiply by the initial investment, [tex]\(P\)[/tex]:

[tex]\[ A(9) = 200 \times 1.566298 = 313.3696 \][/tex]

Rounding this to the nearest cent, we get:

[tex]\[ \$313.37 \][/tex]

Therefore, the correct answer is:

A. [tex]\(\$313.37\)[/tex]