Let's solve the problem of determining which investment option results in more total interest for James.
### Given Data:
- Principal amount [tex]\( P = \$8,000 \)[/tex]
- Simple interest rate [tex]\( r_{\text{simple}} = 7.2\% = 0.072 \)[/tex]
- Continuous compounding interest rate [tex]\( r_{\text{continuous}} = 7\% = 0.07 \)[/tex]
- Time [tex]\( t = 30 \)[/tex] years
### Simple Interest Calculation:
The formula for calculating simple interest is:
[tex]\[ I_{\text{simple}} = P \times r_{\text{simple}} \times t \][/tex]
Substituting the given values into the formula:
[tex]\[ I_{\text{simple}} = 8000 \times 0.072 \times 30 \][/tex]
[tex]\[ I_{\text{simple}} = 8000 \times 2.16 \][/tex]
[tex]\[ I_{\text{simple}} = 17280 \][/tex]
The total interest earned with a simple interest rate of [tex]\(7.2\%\)[/tex] over 30 years is \[tex]$17,280.
### Continuous Compounding Interest Calculation:
The formula for calculating the amount with continuous compounding interest is:
\[ A_{\text{continuous}} = P \times e^{(r_{\text{continuous}} \times t)} \]
Where \( e \) is the base of the natural logarithm (approximately \(2.71828\)).
Substituting the given values into the formula:
\[ A_{\text{continuous}} = 8000 \times e^{(0.07 \times 30)} \]
Calculate the exponent:
\[ 0.07 \times 30 = 2.1 \]
Now, calculate \( e^{2.1} \):
\[ e^{2.1} \approx 8.16617 \]
So:
\[ A_{\text{continuous}} = 8000 \times 8.16617 \]
\[ A_{\text{continuous}} \approx 65329.36 \]
To find the interest earned:
\[ I_{\text{continuous}} = A_{\text{continuous}} - P \]
\[ I_{\text{continuous}} \approx 65329.36 - 8000 \]
\[ I_{\text{continuous}} \approx 57329.36 \]
The total interest earned with continuous compounding interest of \(7\%\) over 30 years is approximately \$[/tex]57,329.36.
### Comparison of Total Interest:
- Simple Interest: \[tex]$17,280
- Continuous Compounding Interest: \$[/tex]57,329.36
Thus, comparing the two amounts, we see that the interest earned from continuous compounding (\[tex]$57,329.36) is significantly higher than that earned from simple interest (\$[/tex]17,280).
### Conclusion:
Investing the \$8,000 at a [tex]\(7\%\)[/tex] interest rate compounded continuously for 30 years results in more total interest than investing at a [tex]\(7.2\%\)[/tex] simple interest rate for the same period.
