find the future value and interest earned if $8704.56 is invested for 7 years at 6% compounded (a)…

find the future value and interest earned if $8704.56 is invested for 7 years at 6% compounded (a) semiannually and (b) continuously.\n(a) the future value when interest is compounded semiannually is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\nthe interest earned is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\n(b) the future value when interest is compounded continuously is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\nthe interest earned is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)

find the future value and interest earned if $8704.56 is invested for 7 years at 6% compounded (a) semiannually and (b) continuously.\n(a) the future value when interest is compounded semiannually is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\nthe interest earned is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\n(b) the future value when interest is compounded continuously is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)\nthe interest earned is approximately $. (type an integer or decimal rounded to the nearest hundredth as needed.)

Answer

Explanation:

Step1: Identify formula for compound - interest (semi - annually)

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 interest is compounded per year, and $t$ is the number of years. Given $P=$8704.56$, $r = 0.06$, $n = 2$ (semi - annually), and $t = 7$.

Step2: Calculate the future value (semi - annually)

$A=8704.56(1 +\frac{0.06}{2})^{2\times7}=8704.56(1 + 0.03)^{14}$. First, calculate $(1 + 0.03)^{14}\approx1.5125897$. Then, $A = 8704.56\times1.5125897\approx13161.97$.

Step3: Calculate the interest earned (semi - annually)

$I=A - P$. So, $I=13161.97-8704.56 = 4457.41$.

Step4: 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. Given $P = 8704.56$, $r=0.06$, and $t = 7$.

Step5: Calculate the future value (continuously)

$A=8704.56\times e^{0.06\times7}=8704.56\times e^{0.42}$. Since $e^{0.42}\approx1.521962$, then $A=8704.56\times1.521962\approx13243.77$.

Step6: Calculate the interest earned (continuously)

$I=A - P$. So, $I=13243.77-8704.56 = 4539.21$.

Answer:

(a) The future value when interest is compounded semiannually is approximately $$13161.97$. The interest earned is approximately $$4457.41$. (b) The future value when interest is compounded continuously is approximately $$13243.77$. The interest earned is approximately $$4539.21$.