Step 1 :We are given a problem about exponential growth. The formula for exponential growth is: \(N = N_0 * 2^{(t/T)}\) where: N is the final amount of bacteria, N_0 is the initial amount of bacteria, t is the time elapsed, and T is the doubling time.
Step 2 :We need to solve for t, so we can rearrange the formula to: \(t = T * log2(N/N_0)\)
Step 3 :We know that \(N_0 = 600\), \(N = 6000\), and \(T = 0.5\) hours (30 minutes). We can plug these values into the formula and solve for t.
Step 4 :Substituting the given values into the formula, we get \(t = 0.5 * log2(6000/600)\)
Step 5 :Solving the equation, we get \(t = 1.660964047443681\)
Step 6 :Rounding to one decimal place, we get \(t = 1.7\)
Step 7 :Final Answer: It will take approximately \(\boxed{1.7}\) hours for the colony to contain 6,000 bacteria.