The growth of Pseudomonas bacteria is modeled as a first-order process with K=0.035 min ^-1 at 37 degrees C. The initial Pseudomonas population density is 1.0 x 10^3 cells/L.
Enter your answers in scientific notations.
A) What is the population density after 4.0 h
B) What is the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L?
A) After 4.0 hours, the population density can be calculated using the formula:
N(t) = N0 * e^(kt)
where N(t) is the population density at time t, N0 is the initial population density, k is the growth rate constant, and e is the base of the natural logarithm.
Given:
N0 = 1.0 x 10^3 cells/L
k = 0.035 min^-1
t = 4.0 h = 240 min
Plugging in the values, we get:
N(240) = (1.0 x 10^3) * e^(0.035 * 240)
Using a calculator, we find that N(240) is approximately 1.52828 x 10^13 cells/L.
So, the population density after 4.0 hours is 1.52828 x 10^13 cells/L.
B) To find the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L, we can rearrange the formula:
N(t) = N0 * e^(kt)
to solve for time t:
t = (ln(N(t)) - ln(N0)) / k
Given:
N(t) = 3.0 x 10^3 cells/L
N0 = 1.0 x 10^3 cells/L
k = 0.035 min^-1
Plugging in the values, we get:
t = (ln(3.0 x 10^3) - ln(1.0 x 10^3)) / 0.035
Using a calculator, we find that t is approximately 1090.592 minutes.
Converting minutes to hours, we have:
t = 1090.592 / 60 ≈ 18.17653 hours
So, the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L is approximately 18.18 hours.
A) To find the population density after 4.0 hours, we can use the exponential growth equation:
N(t) = N₀ * e^(kt)
Where:
N(t) is the population density at time t
N₀ is the initial population density
k is the growth rate constant
t is the time
Given:
N₀ = 1.0 x 10^3 cells/L
k = 0.035 min^-1
First, we need to convert 4.0 hours to minutes:
4.0 hours * 60 min/hour = 240 minutes
Now we can plug in the values:
N(t) = N₀ * e^(kt)
N(t) = (1.0 x 10^3 cells/L) * e^(0.035 min^-1 * 240 min)
Calculating this expression will give us the population density after 4.0 hours.
B) To find the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L, we need to rearrange the exponential growth equation:
N(t) = N₀ * e^(kt)
We know the initial population density (N₀ = 1.0 x 10^3 cells/L) and the final population density (N(t) = 3.0 x 10^3 cells/L). We can solve for t:
t = ln(N(t) / N₀) / k
Substituting the values:
t = ln((3.0 x 10^3 cells/L) / (1.0 x 10^3 cells/L)) / 0.035 min^-1
Calculating this expression will give us the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L.
To solve these questions, we can use the first-order growth equation, which is expressed as:
N(t) = N(0) * e^(kt)
where:
N(t) is the population density at time t,
N(0) is the initial population density,
k is the growth rate constant, and
t is the time in minutes.
Let's solve each question separately:
A) What is the population density after 4.0 h?
First, we need to convert 4.0 hours into minutes. Since there are 60 minutes in an hour, we have:
4.0 hours * 60 minutes/hour = 240 minutes
Now, substitute the given values into the equation, with N(0) = 1.0 x 10^3 cells/L, k = 0.035 min^-1, and t = 240 minutes:
N(t) = 1.0 x 10^3 * e^(0.035 * 240)
Calculating this expression will give us the population density after 4.0 hours.
B) What is the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L?
We can rearrange the growth equation to solve for t:
t = ln(N(t) / N(0)) / k
Substitute the given values into the equation, with N(t) = 3.0 x 10^3 cells/L, N(0) = 1.0 x 10^3 cells/L, and k = 0.035 min^-1:
t = ln(3.0 x 10^3 / 1.0 x 10^3) / 0.035
Calculating this expression will give us the time required for the population to go from 1.0 x 10^3 to 3.0 x 10^3 cells/L.
Remember to use scientific notation to present your answers.