Sure, let's solve this step by step using the continuous compound interest formula. The continuous compound interest formula is given by:
[tex]\[ A = P \times e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the final amount after time [tex]\( t \)[/tex]
- [tex]\( P \)[/tex] is the principal amount (initial investment)
- [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 values:
- [tex]\( A = 19,107 \)[/tex] (final amount)
- [tex]\( P = 12,800 \)[/tex] (principal)
- [tex]\( t = 72 \)[/tex] months. Since [tex]\( t \)[/tex] has to be in years, we convert months to years: [tex]\( t = \frac{72}{12} = 6 \)[/tex] years
We need to find the annual interest rate [tex]\( r \)[/tex]. To do this, let's manipulate the formula to solve for [tex]\( r \)[/tex]:
1. Start with the continuous compound interest formula:
[tex]\[ A = P \times e^{rt} \][/tex]
2. Divide both sides by [tex]\( P \)[/tex]:
[tex]\[ \frac{A}{P} = e^{rt} \][/tex]
3. Take the natural logarithm (ln) of both sides to isolate [tex]\( rt \)[/tex]:
[tex]\[ \ln\left(\frac{A}{P}\right) = rt \][/tex]
4. Solve for [tex]\( r \)[/tex] by dividing both sides by [tex]\( t \)[/tex]:
[tex]\[ r = \frac{\ln\left(\frac{A}{P}\right)}{t} \][/tex]
Now substitute the known values into the equation:
1. Calculate [tex]\( \frac{A}{P} \)[/tex]:
[tex]\[ \frac{19,107}{12,800} \approx 1.492734375 \][/tex]
2. Take the natural logarithm of this value:
[tex]\[ \ln(1.492734375) \approx 0.40060958971176 \][/tex]
3. Divide by [tex]\( t \)[/tex]:
[tex]\[ r = \frac{\ln(1.492734375)}{6} \approx \frac{0.40060958971176}{6} \approx 0.06676826485529448 \][/tex]
Thus, the annual interest rate [tex]\( r \)[/tex] is approximately 0.066768 (as a decimal).
To express [tex]\( r \)[/tex] as a percentage, we multiply by 100:
[tex]\[ r \approx 0.066768 \times 100 = 6.676826485529448\% \][/tex]
Rounded to three decimal places:
[tex]\[ r \approx 6.677\% \][/tex]
So, the interest rate is approximately 6.677%.
