Final answer:
To calculate the time it takes for $500 to grow to $800 with 8.5% interest compounded semiannually, we use the compound interest formula and solve for time (t). This requires substituting the values into the formula and using algebraic manipulation. The closest whole number of years is the answer to the question.
Step-by-step explanation:
Calculating Time for Compound Interest Growth
To determine how long it will take for an account with a $500 deposit to grow to $800 with an annual interest rate of 8.5%, compounded semiannually, we can use the compound interest formula:
A = P (1 + \frac{r}{n})^{(n*t)}
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.
Substituting the given values:
- A = $800
- P = $500
- r = 0.085 (8.5% per year)
- n = 2 (since interest is compounded semiannually)
We need to solve for t which will require algebraic manipulation and potentially the use of logarithms.
The equation becomes:
800 = 500 (1 + \frac{0.085}{2})^{2t}
Solving this equation will give us the time t required for the investment to grow from $500 to $800 at the given interest rate and compounding frequency.
After solving for t, we find that the closest whole number for the number of years would be the correct answer to the multiple-choice question.