interest is compounded semiannually. find the amount in the account after the given time.\nprincipal rate of…

Explanation:

Step1: Identify the compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal form), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years.

Step2: Convert the given values to the appropriate form

Given $P = 4000$, $r=0.06$ (since $6%=0.06$), $n = 2$ (compounded semiannually), and $t = 3$.

Step3: Substitute the values into the formula

$A=4000(1 +\frac{0.06}{2})^{2\times3}=4000(1 + 0.03)^{6}$.

Step4: Calculate the value inside the parentheses

First, calculate $(1 + 0.03)^{6}$. Using the exponentiation operation, $(1.03)^{6}=1.03\times1.03\times1.03\times1.03\times1.03\times1.03\approx1.194052$.

Step5: Calculate the final amount

Then, $A = 4000\times1.194052=4776.208$.

Step6: Round to the nearest cent

Rounding $4776.208$ to the nearest cent gives $4776.21$.

Answer:

$4776.21$