how much money should be deposited today in an account that earns 3% 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 $A = 9000$, $r=0.03$ (since $3%=0.03$), $n = 2$ (compounded semiannually), and $t = 3$.

Step3: Substitute the values into the formula

$P=\frac{9000}{(1 +\frac{0.03}{2})^{2\times3}}=\frac{9000}{(1 + 0.015)^{6}}$. First, calculate $(1 + 0.015)^{6}$. Using the formula $(a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}$, or simply using a calculator, $(1.015)^{6}\approx1.093443$. Then, $P=\frac{9000}{1.093443}\approx8230.84$.

Answer:

$8230.84$