To determine the future value of an investment with continuous compounding, we use the formula:
[tex]\[ A(t) = P \cdot e^{rt} \][/tex]
Given:
- [tex]\( P = 300 \)[/tex] dollars (the principal amount)
- [tex]\( r = 0.07 \)[/tex] (the annual interest rate as a decimal)
- [tex]\( t = 4 \)[/tex] years
Let's substitute these values into the formula:
[tex]\[ A(4) = 300 \cdot e^{0.07 \cdot 4} \][/tex]
First, we need to compute the exponent:
[tex]\[ 0.07 \cdot 4 = 0.28 \][/tex]
Next, we calculate [tex]\( e^{0.28} \)[/tex]. Using the value found for [tex]\( e^{0.28} \)[/tex], we find:
[tex]\[ e^{0.28} \approx 1.323129812 \][/tex]
Now, we multiply this result by the principal amount:
[tex]\[ A(4) = 300 \cdot 1.323129812 \][/tex]
[tex]\[ A(4) \approx 396.9389437 \][/tex]
Finally, we round the result to the nearest cent:
[tex]\[ A(4) \approx 396.94 \][/tex]
Therefore, the future value of the investment after 4 years, when rounded to the nearest cent, is:
[tex]\[ \boxed{396.94} \][/tex]
So, the correct answer is:
B. \$396.93
