Answer:
The investment at 7.42%; $12 599.84
Step-by-step explanation:
The formula for the amount (A) accrued on an investment earning compound interest is
[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]
where
P = the amount of money invested (the principal)
r = the annual interest rate expressed as a decimal fraction
t = the time in years
n = the number of compounding periods per year
1. Daily compounding
Data:
P = $6000
r = 7.42 % = 0.0742
t = 10 yr
n = 365 /yr
Calculation:
[tex]\begin{array}{rcl}\\A& =& P \left (1 + \dfrac{r}{n} \right )^{nt}\\\\& =& 6000 \left(1 + \dfrac{0.0742}{365} \right)^{365\times10}\\\\& =& 6000 \left(1 + 0.0002032888 \right)^{365\times10}\\& =& 6000 (1.000203288)^{3650}\\& =& 6000 \times 2.0999732\\& =& \mathbf{12599.84}\\\end{array}\\[/tex]
2. Quarterly compounding
Data:
r = 7.5 % = 0.075
n = 4 /yr
Calculation:
[tex]\begin{array}{rcl}A& =& P \left (1 + \dfrac{r}{n} \right )^{nt}\\& =& 6000 \left(1 + \dfrac{0.075}{4} \right )^{4\times10}\\& =& 6000 (1 + 0.01875)^{40}\\& =& 6000 (1.01875)^{40}\\& =& 6000 \times 2.102349277\\& =& \mathbf{12614.10}\\\end{array}\\[/tex]
This investment yields the greater return, but the difference is less than $1.50 per year.
3. $ 6000 at 7.42 % compounded daily
From Part 1, the value of the investment is $12 599.84.
