Final answer:
Using the compound interest formula and logarithms to solve for the time, it will take approximately 11.2 years for a $3,300 investment at an annual interest rate of 6.75% to grow to $10,700.
Step-by-step explanation:
To determine how long it will take for the account value to reach $10,700, we can use the formula for compound interest which is 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 in years.Since the interest is compounded annually, n will be 1. Therefore, the formula simplifies to A = P(1 + r)t. Plugging in the values, we want to solve $10,700 = $3,300(1 + 0.0675) for t.
To find out how long it will take for the account value to reach $10,700, we need to determine the number of years it will take for the initial amount to grow to that value. We can use the compound interest formula:A = P(1 + r/n)^(nt)Where:A = final amount ($10,700 in this caseP = principal amount (initial investment of $3,300r = annual interest rate (6.75%n = number of times the interest is compounded per year (usually once per year)t = number of yearsUsing logarithms to solve for t, we get t = log($10,700/$3,300) / log(1 + 0.0675). After calculation, we find that t ≈ 11.2 years. Therefore, it will take approximately 11.2 years for the account value to reach $10,700.