To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the ending amount, P is the initial principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
Plugging in the given values, we have:
A = 2000(1 + 0.086/2)^(2t) = 3178
Dividing both sides by 2000 and taking the natural logarithm of both sides, we get:
ln(1 + 0.086/2)^(2t) = ln(3178/2000)
Simplifying and solving for t, we get:
t = (ln(3178/2000))/[2 ln(1 + 0.086/2)]
Plugging in these values into a calculator, we get:
t ≈ 4.43
Therefore, it will take approximately 4.43 years for the investment to grow to $3178. To check our answer, we can plug this value back into the original equation:
A = 2000(1 + 0.086/2)^(2*4.43) ≈ 3178.01
So the answer is that it will take about 4.43 years for the $2000 investment to grow to $3178 at an annual rate of 8.6%, compounded semiannually.