Final answer:
Using the compound interest formula, we find that a $6000 investment will take approximately 3 years to grow to $6924 at an annual rate of 9.8%, compounded semiannually.
Step-by-step explanation:
To determine how long it will take for a $6000 investment to grow to $6924 at an annual rate of 9.8%, compounded semiannually, we use the compound interest formula:
A = P(1 + 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 or borrowed for, in years.
Given:
- A = $6924
- P = $6000
- r = 9.8% or 0.098
- n = 2 (since the interest is compounded semiannually)
Plugging in these values, we solve for t:
$6924 = $6000(1 + 0.098/2)(2t)
After some calculations, we find:
1.154 = (1 + 0.049)(2t)
t = ln(1.154) / (2 * ln(1.049))
After calculating using a calculator, we find that t is approximately 3 years.
Therefore, it will take approximately 3 years for a $6000 investment to grow to $6924 at an annual rate of 9.8%, compounded semiannually.