To find out how long it will take for [tex]$1900 to grow to $[/tex]3400 at a continuous compound interest rate of 7.9%, we need to use the formula for continuous compound interest, which is:
[tex]\[ A = Pe^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount the principal will grow to after time [tex]\( t \)[/tex],
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form),
- [tex]\( t \)[/tex] is the time in years, and
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
In this scenario, we are given:
- [tex]\( A = \$3400 \)[/tex],
- [tex]\( P = \$1900 \)[/tex],
- [tex]\( r = 7.9\% = 0.079 \)[/tex], as a decimal.
We are looking to find [tex]\( t \)[/tex], the number of years it takes for the principal to grow to the final amount. We can rearrange the formula to solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(A/P)}{r} \][/tex]
Using the given values, we find:
[tex]\[ t = \frac{\ln(3400/1900)}{0.079} \][/tex]
Following through with this calculation, we'll reach the conclusion that [tex]\( t \approx 7.366095512021785 \)[/tex] years when not rounded. When we round this value to the nearest hundredth, we get:
[tex]\[ t \approx 7.37 \][/tex] years.
Based on the options provided, the correct answer is B. 7.37 years.
