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 $P$ is the principal amount (initial deposit), $A$ is the future value, $r$ 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 given values to appropriate form

Given $A=$15000$, $r = 0.08$ (since $8%=0.08$), $n = 2$ (compounded semiannually), and $t = 3$ years.

Step3: Substitute values into the formula

$P=\frac{15000}{(1 +\frac{0.08}{2})^{2\times3}}=\frac{15000}{(1 + 0.04)^{6}}$.

Step4: Calculate the denominator

$(1 + 0.04)^{6}=1.04^{6}\approx1.265319018$.

Step5: Calculate the present value

$P=\frac{15000}{1.265319018}\approx11854.87$.

Answer:

$11854.87$