find the future value and interest earned if $8904.56 is invested for 8 years at 5% compounded (a)…

Explanation:

Step1: Identify compound - interest formula for semi - annual compounding

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times compounded per year, and $t$ is the number of years. Here, $P=$8904.56$, $r = 0.05$, $n = 2$ (semi - annual), and $t = 8$. $A=8904.56(1 +\frac{0.05}{2})^{2\times8}$

Step2: Calculate the future value for semi - annual compounding

First, calculate the value inside the parentheses: $1+\frac{0.05}{2}=1 + 0.025=1.025$. Then, calculate the exponent: $2\times8 = 16$. So, $A=8904.56\times(1.025)^{16}$. Using a calculator, $(1.025)^{16}\approx1.484505$. Then $A = 8904.56\times1.484505\approx13215.77$. The interest earned $I=A - P=13215.77-8904.56 = 4311.21$.

Step3: Identify formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the number of years. Here, $P = 8904.56$, $r=0.05$, and $t = 8$. $A=8904.56\times e^{0.05\times8}$

Step4: Calculate the future value for continuous compounding

First, calculate the exponent: $0.05\times8 = 0.4$. Then, $e^{0.4}\approx1.491825$. So, $A=8904.56\times1.491825\approx13283.77$. The interest earned $I=A - P=13283.77-8904.56 = 4379.21$.

Answer:

(a) The future value when interest is compounded semiannually is approximately $$13215.77$. The interest earned is approximately $$4311.21$. (b) The future value when interest is compounded continuously is approximately $$13283.77$. The interest earned is approximately $$4379.21$.