To determine how much an investment of \[tex]$570 will be worth in 10 years with a continuous compound interest rate of 4% per year, we use the continuous compound interest formula:
\[ A = P e^{rt} \]
where:
- \( P \) is the principal amount (initial investment), which is \$[/tex]570.
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal), which is 0.04 (since 4% = 4/100 = 0.04).
- [tex]\( t \)[/tex] is the time the money is invested for, which is 10 years.
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).
We can substitute the values into the formula:
[tex]\[ A = 570 \cdot e^{0.04 \cdot 10} \][/tex]
First, we calculate the exponent:
[tex]\[ 0.04 \cdot 10 = 0.4 \][/tex]
Next, we compute [tex]\( e^{0.4} \)[/tex].
After finding the value of [tex]\( e^{0.4} \)[/tex], we then multiply this by the principal amount:
[tex]\[ A = 570 \cdot e^{0.4} \][/tex]
Finally, we get the approximate value for [tex]\( A \)[/tex]. When computed, the final value is approximately \[tex]$850.34.
Therefore, the investment will be worth approximately \$[/tex]850.34 in 10 years. The correct answer from the options given is:
[tex]\[
\$ 850.34
\][/tex]
