how much money should be deposited today in an account that earns 8% compounded semiannually so that it will…

Explanation:

Step1: Identify the compound - interest formula

The compound - interest formula for present value is $P = \frac{A}{(1+\frac{r}{n})^{nt}}$, where $A$ is the future value, $P$ is the present value (the amount to be deposited), $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years.

Step2: Convert the given values to the appropriate form

We have $A=$13000$, $r = 0.08$ (since $8%=0.08$), $n = 2$ (compounded semiannually), and $t = 3$ years.

Step3: Substitute the values into the formula

$P=\frac{13000}{(1 +\frac{0.08}{2})^{2\times3}}=\frac{13000}{(1 + 0.04)^{6}}$. First, calculate $(1 + 0.04)^{6}$. Using the formula $a^n$, where $a = 1.04$ and $n = 6$, we get $(1.04)^{6}=1.04\times1.04\times1.04\times1.04\times1.04\times1.04\approx1.265319$. Then, $P=\frac{13000}{1.265319}\approx10273.97$.

Answer:

$10273.97$