The doubling function (y = y base0 2^(1/D)) can be used to model exponential growth when the doubling time is D. The bacterium Escherichia coli has a doubling period of 0.32 h. A culture of E. coli starts with 100 bacteria.
a) Determine the equation for the number of bacteria, y, in x hours.
b) Graph your equation
c) Graph the inverse
d) Determine the equation of the inverse. What does this equation represent?
e) How many hours will it take for there to be 450 bacteria in the culture? Explain your strategy.
equation:
y = 100 (2)^(t/3.2)
..
..
to get 450 bacteria
450 = 100 (2)^(x/3.2)
4.5 = 2^(x/3.2)
take log of both sides and use log rules
(x/3.2)log 2 = log 4.5
x/3.2 = log 4.5/log2
x = 6.944 hrs. or appr 7 hours
a) To determine the equation for the number of bacteria, y, in x hours, we can use the doubling function:
y = y₀ * 2^(x/D)
where:
- y is the number of bacteria at time x
- y₀ is the initial number of bacteria (100 in this case)
- x is the time in hours
- D is the doubling period (0.32 hours in this case)
So, the equation for the number of bacteria, y, in x hours is:
y = 100 * 2^(x/0.32)
b) To graph the equation, we can plot the number of bacteria (y) on the y-axis and time (x) on the x-axis.
c) To graph the inverse, we need to swap the x and y values. In this case, we need to plot time (x) on the y-axis and the number of bacteria (y) on the x-axis.
d) The equation of the inverse can be found by swapping x and y in the original equation:
x = 100 * 2^(y/0.32)
Simplifying, we get:
y = 0.32 * log₂(x/100)
This equation represents the time it takes for the number of bacteria to reach a specific value, x.
e) To calculate the number of hours it will take for there to be 450 bacteria in the culture, we can rearrange the original equation:
450 = 100 * 2^(x/0.32)
Dividing both sides by 100:
4.5 = 2^(x/0.32)
Taking the logarithm of both sides:
log₂(4.5) = x/0.32
Now, solving for x:
x = 0.32 * log₂(4.5)
To calculate this value, you can use a calculator or software.
a) To determine the equation for the number of bacteria, y, in x hours, we can use the doubling function. Since the doubling time is D = 0.32 h and the initial number of bacteria is 100, we can substitute these values into the doubling function formula:
y = y₀ * 2^(x/D)
Substituting y₀ = 100 and D = 0.32, we get:
y = 100 * 2^(x/0.32)
Therefore, the equation for the number of bacteria, y, in x hours is y = 100 * 2^(x/0.32).
b) To graph the equation, we can plot the values of x and y on a graph. Choose a range of x values and calculate the corresponding y values using the equation. Then, plot the points (x, y) on a graph and connect them to visualize the relationship between x and y.
c) To graph the inverse, we need to swap the x and y values from the original equation. The inverse equation is:
x = 100 * 2^(y/0.32)
Similarly, choose a range of y values and calculate the corresponding x values using this inverse equation. Plot the points (y, x) on a graph and connect them.
d) To determine the equation of the inverse, we need to solve the inverse equation for y. Rearranging the inverse equation, we get:
2^(y/0.32) = x / 100
Taking the logarithm of both sides with base 2, we have:
log₂(2^(y/0.32)) = log₂(x / 100)
(y/0.32) = log₂(x / 100)
Multiplying both sides by 0.32, we get:
y = 0.32 * log₂(x / 100)
Therefore, the equation of the inverse is y = 0.32 * log₂(x / 100). This equation represents the doubling time it takes for the number of bacteria to reach a certain level x.
e) To calculate how many hours it will take for there to be 450 bacteria in the culture, we can use the original equation y = 100 * 2^(x/0.32). Substitute y = 450 into the equation and solve for x:
450 = 100 * 2^(x/0.32)
Divide both sides by 100:
4.5 = 2^(x/0.32)
To solve for x, we can take the logarithm of both sides with base 2:
log₂(4.5) = (x/0.32)
Using a calculator to evaluate log₂(4.5), we find it is approximately 2.1699.
Now, multiply both sides by 0.32:
0.32 * log₂(4.5) = x
x is approximately equal to 0.6944.
Therefore, it will take approximately 0.6944 hours for there to be 450 bacteria in the culture.