Final answer:
Using the half-life decay model, we solved for the decay constant k and then used it to calculate the time t when only 50 bacteria would remain. After determining k, we plugged the values into the formula, took natural logarithms, and solved for t, rounding to the nearest whole number as the final step.
Step-by-step explanation:
To determine after how many minutes there will be 50 bacteria remaining, given that the sample starts with 600 bacteria and after 12 minutes there are 480 bacteria, we can use the half-life decay model equation N(t) = N0 * e^(kt), where N(t) is the quantity of substance that still remains and has not yet decayed after t time, N0 is the initial quantity of substance, e is Euler's number (approx. 2.71828), and k is the decay constant.
First, we need to find the decay constant k. Since we know that N0 is 600 and N(t) is 480 after 12 minutes, we can set up the equation 480 = 600 * e^(12k) and solve for k. This yields:
\( e^(12k) = \frac{480}{600} = 0.8 \).
Next, we take the natural logarithm of both sides to solve for k:
\( 12k = ln(0.8) \)
\( k = \frac{ln(0.8)}{12} \)
After calculating and rounding k to four decimal places, we find that:
\( k = -0.0183 \).
We now use the decay constant to find the time t when there are 50 bacteria left:
\( 50 = 600 * e^(-0.0183t) \).
Then we take the natural logarithm and solve for t:
\( ln(50/600) = -0.0183t \)
\( t = \frac{ln(50/600)}{-0.0183} \)
After calculating the time t, we round it to the nearest whole number to get the final answer.