Lauren can withdraw from the account for approximately 2 years and 1 month.
To find out how long Lauren can withdraw from the account, we need to use the formula for compound interest: 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, the principal amount is $10,000, the annual interest rate is 9.6%, and interest is compounded semi-annually, so n = 2.
We want to find t when the value of A is $500. Rearranging the formula, we have t = log(A/P) / (n * log(1 + r/n)). Plugging in the values, we get t = log(500/10000) / (2 * log(1 + 0.096/2)).
Calculating this using a calculator, we find t to be approximately 25.39 months, which is equal to 2 years and 1 month (or 25 months) rounded down.