Answer
a) 5026.16 dollars
b) 5634.10 dollars
c) 6315.58 dollars
Work Step by Step
First one has to find the function that models the example. This uses the compound interest function $A(t)=P\left(1+\frac{r}{n}\right)^{nt}$, where $P$ is the principal value, $r$ is the interest rate per year, $n$ is the number of times the interest is compounded per year, and $t$ is the number of years.
The principal $p$ is \$4000 dollars, the interest rate $r$ is 5.75% or 0.0575 in decimal form, and the number of times it is compounded annually $n$ is 4. Now the function can be described:
$A(t)=4000\cdot \left(1+\frac{0.0575}{4}\right)^{4t}$
Now one only needs to calculate $A(4)$, $A(6)$, and $A(8)$ which corresponds to the investment after 4, 6, and 8 years, respectively.
$A(4)=4000\cdot \left(1.014375\right)^{4(4)}=$
$4000\cdot (1.014375)^{16}=5026.16$ dollars
$A(6)=4000\cdot \left(1.014375\right)^{4(6)}=$
$4000\cdot (1.014375)^{24}=5634.10$ dollars
$A(8)=4000\cdot \left(1.014375\right)^{4(8)}=$
$4000\cdot (1.014375)^{32}=6315.58$ dollars
