how much money should be deposited today in an account that earns 4% compounded semiannually so that it will…

Explanation:

Step1: Identify the compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the future value, $P$ is the principal (initial deposit), $r$ 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. We want to solve for $P$, so we can rewrite the formula as $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$.

Step2: Convert the given values to the appropriate form

The annual interest rate $r = 4%=0.04$, the number of times compounded per year $n = 2$ (since it's compounded semiannually), the number of years $t = 3$, and the future value $A=$11000$.

Step3: Substitute the values into the formula

$P=\frac{11000}{(1+\frac{0.04}{2})^{2\times3}}$. First, calculate the value inside the parentheses: $\frac{0.04}{2}=0.02$, and $1 + 0.02=1.02$. Then, calculate the exponent: $2\times3 = 6$. So, $(1.02)^{6}\approx1.126162$.

Step4: Calculate the principal $P$

$P=\frac{11000}{1.126162}\approx9767.67$.

Answer:

$9767.67$