Certainly! Let's solve this step-by-step using the continuous compound interest formula:
The continuous compound interest formula is given by:
[tex]\[ A = P e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( t \)[/tex] is the time the money is invested for in years.
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Given values:
- [tex]\( P = 740 \)[/tex] dollars
- [tex]\( r = 0.11 \)[/tex] (since 11% = 0.11)
- [tex]\( t = 7 \)[/tex] years
Let's calculate:
1. Identify the values:
- Principal (P) = [tex]$740$[/tex]
- Annual interest rate (r) = [tex]$0.11$[/tex]
- Time (t) = [tex]$7$[/tex] years
2. Substitute the values into the formula:
[tex]\[ A = 740 \times e^{0.11 \times 7} \][/tex]
3. Calculate the exponent part:
[tex]\[ 0.11 \times 7 = 0.77 \][/tex]
4. Using the value of e (approximately 2.71828), calculate:
[tex]\[ e^{0.77} \approx 2.157 \][/tex]
5. Now multiply the principal amount by this value:
[tex]\[ A = 740 \times 2.157 \][/tex]
[tex]\[ A \approx 1598.23 \][/tex]
So, after 7 years, the investment will be worth approximately \[tex]$1598.23.
Thus, the correct choice based on the given options would be the value closest to $[/tex]\[tex]$ 1598$[/tex], which matches [tex]$\$[/tex] 1,598$.
