Answer:
To determine how long it would take for the account balance to reach $20,000 with an initial investment of $10,000 and an annual interest rate of 6% compounded annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final account balance
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, we have:
P = $10,000
A = $20,000
r = 6% or 0.06 (as a decimal)
n = 1 (compounded annually)
Plugging in the values, we have:
$20,000 = $10,000(1 + 0.06/1)^(1*t)
Simplifying the equation:
2 = (1 + 0.06)^t
Taking the natural logarithm of both sides:
ln(2) = ln((1 + 0.06)^t)
Using the logarithmic property, we can bring down the exponent:
ln(2) = t * ln(1 + 0.06)
Dividing both sides by ln(1 + 0.06):
t = ln(2) / ln(1 + 0.06)
Using a calculator, we can find:
t ≈ 11.896
Therefore, it would take approximately 11.896 years for the account balance to reach $20,000 with a $10,000 initial investment and a 6% annual interest rate compounded annually.