Problem

A bank features a savings account that has an annual percentage rate of $r=4.3 \%$ with interest compounded semi-annually. Melynda deposits $\$ 11,000$ into the account.

The account balance can be modeled by the exponential formula $S(t)=P\left(1+\frac{r}{n}\right)^{n t}$, where $S$ is the future value, $P$ is the present value, $r$ is the annual percentage rate, $n$ is the number of times each year that the interest is compounded, and $t$ is the time in years.
(A) What values should be used for $P, r$, and $n$ ?
\[
P=\square, \quad r=\square, \quad n=
\]
(B) How much money will Melynda have in the account in 10 years?
\[
\text { Answer }=\$ \text {. }
\]

Round answer to the nearest penny.
(C) What is the annual percentage yield (APY) for the savings account? (The APY is the actual or effective annual percentage rate which includes all compounding in the year).
\[
A P Y=\square \text {. }
\]

Round answer to 3 decimal places.
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Answer

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Answer

Substituting the values of \(r\) and \(n\) into the formula, we calculate \(APY=\left(1+\frac{0.043}{2}\right)^2-1\) and round the answer to 3 decimal places.

Steps

Step 1 :Given that Melynda deposits $11,000 into a savings account with an annual percentage rate of 4.3% compounded semi-annually, we can identify the values for \(P\), \(r\), and \(n\) in the exponential formula \(S(t)=P\left(1+\frac{r}{n}\right)^{n t}\). Here, \(P\) is the present value, \(r\) is the annual percentage rate, \(n\) is the number of times each year that the interest is compounded, and \(t\) is the time in years.

Step 2 :Substituting the given values into the variables, we get \(P=\boxed{11000}\), \(r=\boxed{0.043}\), and \(n=\boxed{2}\).

Step 3 :To find out how much money Melynda will have in the account in 10 years, we substitute the values of \(P\), \(r\), \(n\), and \(t=10\) into the formula and calculate the future value \(S(t)\).

Step 4 :Using Python code, we calculate \(S(t)=11000\left(1+\frac{0.043}{2}\right)^{2*10}\) and round the answer to the nearest penny.

Step 5 :The annual percentage yield (APY) for the savings account can be calculated using the formula \(APY=\left(1+\frac{r}{n}\right)^n-1\).

Step 6 :Substituting the values of \(r\) and \(n\) into the formula, we calculate \(APY=\left(1+\frac{0.043}{2}\right)^2-1\) and round the answer to 3 decimal places.

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