A bacteria culture grows with constant relative growth rate. The bacteria count was 784 after 2 hours and 117649 after 6 hours.
What is the relative growth rate?
What was the initial size of the culture?
Find an expression for the number of bacteria after t hours.
Find the number of cells after 3 hours.
When will the population reach 1000000?
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To find the relative growth rate, we can use the formula:
Relative growth rate (r) = (ln(N2) - ln(N1)) / (t2 - t1)
where N1 and N2 are the initial and final counts respectively, and t1 and t2 are the corresponding time intervals.
Given that the bacteria count was 784 after 2 hours (N1 = 784) and 117649 after 6 hours (N2 = 117649), we can substitute these values into the formula to find the relative growth rate:
r = (ln(117649) - ln(784)) / (6 - 2)
Using a calculator, we can evaluate this expression:
r ≈ (11.676 - 6.664) / 4 ≈ 1.003
Therefore, the relative growth rate is approximately 1.003.
To find the initial size of the culture, we can use the formula:
N0 = N / e^(rt)
where N is the final count, r is the relative growth rate, and t is the time interval.
Substituting the known values N = 784, r = 1.003, and t = 2 hours:
N0 = 784 / e^(1.003 * 2)
Using a calculator, we can evaluate this expression:
N0 ≈ 784 / 7.475 ≈ 104.892
Therefore, the initial size of the culture is approximately 104.892.
An expression for the number of bacteria after t hours can be found using the formula:
N(t) = N0 * e^(rt)
where N(t) is the number of bacteria after t hours, N0 is the initial size of the culture, r is the relative growth rate, and t is the time interval.
Substituting the known values N0 ≈ 104.892 and r ≈ 1.003:
N(t) = 104.892 * e^(1.003 * t)
To find the number of cells after 3 hours, we can substitute t = 3 into the expression:
N(3) = 104.892 * e^(1.003 * 3)
Using a calculator, we can evaluate this expression:
N(3) ≈ 104.892 * e^(1.003 * 3) ≈ 104.892 * e^(3.009) ≈ 1253.09
Therefore, the number of cells after 3 hours is approximately 1253.09.
To find when the population will reach 1000000, we can set N(t) = 1000000 in the expression:
1000000 = 104.892 * e^(1.003 * t)
To solve for t, we divide both sides by 104.892 and take the natural logarithm:
ln(1000000 / 104.892) = 1.003 * t
Using a calculator, we can evaluate this expression:
ln(9531.679) ≈ 1.003 * t
t ≈ ln(9531.679) / 1.003 ≈ 6.006
Therefore, the population will reach 1000000 after approximately 6.006 hours.
To find the relative growth rate, we can use the formula:
Relative Growth Rate (r) = (Final size / Initial size)^(1 / Time) - 1
Using the given information, we can calculate the relative growth rate:
Final size = 117649
Initial size = ?
Time = 6 hours
Plugging the values into the formula, we have:
r = (117649 / Initial size)^(1 / 6) - 1
To find the initial size of the culture, we can rearrange the formula:
Initial size = Final size / (1 + r)^Time
Plugging in the values, we have:
Initial size = 117649 / (1 + r)^6
Now, let's find an expression for the number of bacteria after t hours. Since the culture grows with a constant relative growth rate, we can use the formula:
Number of bacteria = Initial size * (1 + r)^t
Plugging in the values, we have:
Number of bacteria (t) = Initial size * (1 + r)^t
To find the number of cells after 3 hours, we can substitute t = 3 into the expression:
Number of bacteria (3) = Initial size * (1 + r)^3
Finally, to find when the population will reach 1000000, we can set the number of bacteria equal to 1000000 and solve for t:
1000000 = Initial size * (1 + r)^t
Now, let's calculate the relative growth rate, the initial size of the culture, find an expression for the number of bacteria after t hours, the number of cells after 3 hours, and determine when the population will reach 1000000.