Final answer:
To find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 6.5% per year, compounded quarterly, we can use the formula for compound interest. By plugging in the values and solving for t, we find that it will take approximately 4.25 years for the investment to reach $8,000.
Step-by-step explanation:
To find the time required for an investment of $3,000 to grow to $8,000 at an interest rate of 6.5% per year, compounded quarterly, we can use the formula for compound interest.
The formula for compound interest is: A = P(1 + r/n)^(nt), where A is the future value, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.
In this case, we have P = $3,000, A = $8,000, r = 6.5% = 0.065, and n = 4 (quarterly compounding). We need to solve for t.
Plugging in the values, we have: $8,000 = $3,000(1 + 0.065/4)^(4t)
Dividing both sides by $3,000 gives: 8/3 = (1 + 0.065/4)^(4t)
Raising both sides to the power of (4t) gives: (8/3)^(1/4) = 1 + 0.065/4
Subtracting 1 from both sides gives: (8/3)^(1/4) - 1 = 0.065/4
Multiplying both sides by 4 gives: 4[(8/3)^(1/4) - 1] = 0.065
Dividing both sides by 0.065 gives: t = [4/0.065][(8/3)^(1/4) - 1]
Calculating this value gives t = 4.246, which is approximately 4.25 years (rounded to two decimal places).