To determine the account balance after 14 years for Ming's investment, we use the formula for continuous compounding interest:
[tex]\[ A = Pe^{rt} \][/tex]
where:
- [tex]\(A\)[/tex] is the account balance (the amount of money in the account after [tex]\(t\)[/tex] years),
- [tex]\(P\)[/tex] is the principal (the initial amount of money invested),
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal),
- [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:
- [tex]\(P = 3934\)[/tex] dollars,
- [tex]\(r = 0.04\)[/tex],
- [tex]\(t = 14\)[/tex] years.
We can now substitute these values into the formula:
[tex]\[ A = 3934 \times e^{0.04 \times 14} \][/tex]
First, we calculate the exponent:
[tex]\[ 0.04 \times 14 = 0.56 \][/tex]
Next, we find [tex]\( e^{0.56} \)[/tex]:
[tex]\[ e^{0.56} \approx 1.7517 \][/tex]
Now, we substitute this back into the equation:
[tex]\[ A = 3934 \times 1.7517 \][/tex]
Finally, we perform the multiplication:
[tex]\[ A \approx 3934 \times 1.7517 \approx 6890.49 \][/tex]
So, the account balance after 14 years will be approximately $6,890.49.
