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Sagot :
The answer is $3,619.80.
To answer this question, we need to remember that the formula for compound interest is given by:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where:
• A will be the final balance after the given period (accrued amount).
,• P is the Principal (the amount we deposit). In this case, we have $3500.
,• r is the interest rate. In this case, we have 6.75% = 6.75/100.
,• n is the number of times per year compounded. In this case, the number of times is monthly, that is, n = 12.
,• t is the time in years. Since we have here that we need the balance after 6 months, we have that 6 months is 1/2 year = 0.5year.
Then we can substitute the corresponding values into the general equation as follows:
[tex]\begin{gathered} A=P(1+\frac{r}{n})^{nt} \\ A=3500(1+\frac{\frac{6.75}{100}}{12})^{12(\frac{1}{2})} \\ A=3619.79864399 \end{gathered}[/tex]If we round this result to two decimals, we have $3619.80
Using the given equation, we have:
[tex]f(x)=a(1+\frac{r}{n})^{xn}[/tex]Where:
• a is the initial amount. In this case, a = $3500.
,• r is the rate (of interest). In this case, r = 6.75% = 6.75/100.
,• x is the time (in years). Since we have here 6 months, then 6 months is 1/2 year.
,• n is the number of times per year compounded. In this case, the number of times is monthly, that is, n = 12.
Then we can apply the equation as follows:
x = 1/2
[tex]\begin{gathered} f(x)=a(1+\frac{r}{n})^{xn} \\ f(\frac{1}{2})=3500(1+\frac{\frac{6.75}{100}}{12})^{(\frac{1}{2})(12)} \\ f(\frac{1}{2})=3619.79864399 \end{gathered}[/tex]If we round our result to the nearest hundredths, we have:
[tex]f(\frac{1}{2})=\$3,619.80[/tex]We need to remember that:
[tex]6.75\%=\frac{6.75}{100}=0.0675[/tex]And we have that n is the number of times per year compounded. In this case, the number of times is monthly, that is, n = 12. If we have a daily compounded amount, the value for n = 365. If it is quarterly compounded, we have that n = 4, and so on.
We got the same answer in both cases.
In summary, the balance in the account after the given period (6 months) is $3,619.80 (option a) - the interest period was 6 months (1/2 year).
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