The growth rate of Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 30 min.
(b) How long would it take for a colony of 50 cells to increase to a population of 1 million? (Round your answer to the nearest whole number.)
we have already done a half life one but anyway
ds/dt = k s
ds/s = k dt
ln s = k t + c
when t = 0, ln si = c
si is initial size
ln s = k t + ln si
ln (s/si) = k t
s/si = e^kt
2 = e^k(30 min)
ln 2 = 30 k
k = .0231
1,000,000/50 = e^.0231 t
ln (20,000) = .0231 t
t = 428.6 = 429 minutes
about 7 1/4 hours
Thinking of the doubling rate, we can see that after t hours, the population is
P(t) = 50 * 2^(2t) = 50*4^t
So, we want
50*4^t = 1000000
4^t = 20000
t log4 = log 20000
t = log20000/log4 = 7.14 hours
To solve this problem, we can use the exponential growth formula:
N(t) = N0 * 2^(t/T)
where:
N(t) = population at time t
N0 = initial population
t = time
T = doubling time
Given that the initial population (N0) is 50 cells, and the population at time t should be 1 million, we need to find the value of t.
Plug in the values into the formula:
1,000,000 = 50 * 2^(t/30)
Divide both sides of the equation by 50:
20,000 = 2^(t/30)
Now, take the logarithm (base 2) of both sides to solve for (t/30):
log₂(20,000) = t/30
Using a calculator, calculate the logarithm:
log₂(20,000) ≈ 14.2877
Multiply both sides by 30 to solve for t:
t ≈ 14.2877 * 30 ≈ 428.63
Rounding to the nearest whole number, it would take approximately 429 minutes for the colony of 50 cells to increase to a population of 1 million.
To determine how long it would take for a colony of 50 cells to increase to a population of 1 million, we can use the equation for exponential growth:
N = N₀ * 2^(t / T)
Where:
- N is the final population size
- N₀ is the initial population size
- t is the time in minutes
- T is the doubling time in minutes
From the question, we are given that the doubling time (T) for Escherichia coli is approximately 30 minutes.
Let's substitute the known values into the equation:
1,000,000 = 50 * 2^(t / 30)
Now, let's solve for t by isolating the variable:
2^(t / 30) = 1,000,000 / 50
Taking the logarithm of both sides of the equation:
t / 30 = log₂(1,000,000 / 50)
Using a calculator or logarithmic table, calculate the logarithm:
t / 30 = log₂(20,000)
Now, isolate the variable by multiplying both sides by 30:
t = 30 * log₂(20,000)
Using a calculator, solve for t:
t ≈ 133.3
Rounding to the nearest whole number, it would take approximately 133 minutes for a colony of 50 cells to increase to a population of 1 million.