Final answer:
To calculate how long it takes for $800 to grow to $2000 with 12.65% interest compounded monthly, use the compound interest formula. After substituting the given values, solve for time (t) to find the duration needed for the investment to reach the target amount.
Step-by-step explanation:
The question involves the calculation of the time it takes for an investment to grow to a certain amount with compound interest. The formula to use is:
A = P(1 + \frac{r}{n})^{nt}
where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
In this case, to find the time (t) it takes for $800 to grow to $2000 at 12.65% interest compounded monthly:
A = $2000
P = $800
r = 12.65% or 0.1265
n = 12 (since the interest is compounded monthly)
We rearrange the formula to solve for t:
t = \frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})}
Substituting the given values:
t = \frac{\log(\frac{2000}{800})}{12 \cdot \log(1 + \frac{0.1265}{12})}
After calculating, we can approximate the value of t to find out how long it will take for the account balance to grow to $2000. By solving this expression, we can find the correct option from the multiple-choice answers provided.