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Lactobacillus acidophilus is one of the many bacteria used to turn milk into yogurt. The population \( P \) of a colony of 3600 bacteria at time \( t \) minutes can be modeled by the function:

\[ P(t) = 3600 \times 2^{t/73} \]

How long does it take the population (in hours) to reach 1,792,000? You may round to the nearest 5-minute increment. Please show all of your work and submit those with your answer.

Answer :

Final answer:

To find the time it takes for a population of Lactobacillus acidophilus to grow from 3600 to 1,792,000, we solve the given equation for t, leading to a result of approximately 15 minutes or 0.25 hours.

Explanation:

The question asks us to find how long it takes a colony of Lactobacillus acidophilus bacteria, starting with a population of 3600, to reach a population of 1,792,000 using the model P(x) = 3600(2)^t/73, where t is in minutes.

To solve this problem, we first isolate t in the equation and then solve for it using the given final population value.


Step 1: Plug the final population into the model equation.

  • 1,792,000 = 3600(2)^t/73

Step 2: Solve for t.

  • 1,792,000 / 3600 = (2)^t/73
  • 498 = (2)^t/73
  • 498 * 73 = 2^t
  • 36354 = 2^t

To find t, we use logarithms:

  • t = log2(36354)
  • t ≈ 15.14 (minutes)

Since the question asks for the time in hours and rounded to the nearest 5-minute increment, we convert 15.14 minutes to hours and round:

  • 15.14 minutes ≈ 15 minutes = 0.25 hours

So, it takes approximately 0.25 hours, or 15 minutes, for the population of bacteria to reach 1,792,000.

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Rewritten by : Jeany

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