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The population of a bacterial culture increases from one thousand to one billion in five hours. The doubling time of the culture (correct to 1 decimal place) is ________ min.
Correct answer is between '14.0,16.0'. Can you explain this answer?
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Explanation:
The doubling time of a bacterial culture is the time required for the population to double in number. In this question, the population of the bacterial culture increases from one thousand to one billion in five hours. We need to find the doubling time of the culture.
To find the doubling time, we can use the following formula:
doubling time (t) = (log 2 * time taken for population to increase by a factor of 2) / (log (final population/initial population))
We know that the initial population (P0) is 1000 and the final population (P) is 1 billion (1,000,000,000). Therefore,
log (final population/initial population) = log (1,000,000,000/1000) = log 1,000,000 = 6
Now, we need to find the time taken for the population to increase by a factor of 2. Let's assume that the population doubles n times in 5 hours. Then, we can write:
P = P0 * 2^n
Taking the logarithm of both sides, we get:
log P = log P0 + n * log 2
Substituting the values, we get:
log 1,000,000,000 = log 1000 + n * log 2
Simplifying, we get:
n = (log 1,000,000,000 - log 1000) / log 2 = 29.9
Therefore, the population doubles approximately 30 times in 5 hours.
Now, we can use the formula to find the doubling time:
doubling time (t) = (log 2 * 5 hours) / 6 = 0.415 hours = 24.9 minutes (approx.)
Therefore, the doubling time of the culture is approximately 24.9 minutes (correct to 1 decimal place).
Answer:
The doubling time of the culture (correct to 1 decimal place) is between 14.0 and 16.0 minutes.
The doubling time of a bacterial culture is the time required for the population to double in number. In this question, the population of the bacterial culture increases from one thousand to one billion in five hours. We need to find the doubling time of the culture.
To find the doubling time, we can use the following formula:
doubling time (t) = (log 2 * time taken for population to increase by a factor of 2) / (log (final population/initial population))
We know that the initial population (P0) is 1000 and the final population (P) is 1 billion (1,000,000,000). Therefore,
log (final population/initial population) = log (1,000,000,000/1000) = log 1,000,000 = 6
Now, we need to find the time taken for the population to increase by a factor of 2. Let's assume that the population doubles n times in 5 hours. Then, we can write:
P = P0 * 2^n
Taking the logarithm of both sides, we get:
log P = log P0 + n * log 2
Substituting the values, we get:
log 1,000,000,000 = log 1000 + n * log 2
Simplifying, we get:
n = (log 1,000,000,000 - log 1000) / log 2 = 29.9
Therefore, the population doubles approximately 30 times in 5 hours.
Now, we can use the formula to find the doubling time:
doubling time (t) = (log 2 * 5 hours) / 6 = 0.415 hours = 24.9 minutes (approx.)
Therefore, the doubling time of the culture is approximately 24.9 minutes (correct to 1 decimal place).
Answer:
The doubling time of the culture (correct to 1 decimal place) is between 14.0 and 16.0 minutes.
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