A beginning yogurt maker decides to test how consistently his yogurt cultures grow. Hestarts with 105 cells/ml in the cultures. After 4 hours, he has 20 × 105 cells/ml. What is thedoubling time given the amount of cells overtime is 𝐴 = 𝐴0𝑒𝑘𝑡
Question
A beginning yogurt maker decides to test how consistently his yogurt cultures grow. Hestarts with 105 cells/ml in the cultures. After 4 hours, he has 20 × 105 cells/ml. What is thedoubling time given the amount of cells overtime is 𝐴 = 𝐴0𝑒𝑘𝑡
Solution
The doubling time of a population under exponential growth can be calculated using the formula T = ln(2)/k, where k is the growth rate.
First, we need to find the growth rate (k). We can use the formula A = A0 * e^(kt), where A is the final amount, A0 is the initial amount, t is the time, and e is the base of natural logarithms.
Given in the problem, A = 20 * 105 cells/ml, A0 = 105 cells/ml, and t = 4 hours.
We can plug these values into the formula and solve for k:
20 * 105 = 105 * e^(4k)
Divide both sides by 105:
20 = e^(4k)
Take the natural logarithm of both sides:
ln(20) = 4k
Solve for k:
k = ln(20) / 4
Now that we have k, we can find the doubling time (T) using the formula T = ln(2)/k:
T = ln(2) / (ln(20) / 4)
Solve for T to get the doubling time.
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