The bacterium Escherichia coli, or
E-coli, has a doubling period of 0.32 h. If a culture starts with 100 bacteria:
(a) Determine the equation for the number of bacteria, y, in x hours.
b) How many hours will pass before there are 450 bacteria in the culture?
To determine the equation for the number of bacteria, we need to use the concept of exponential growth, where the equation is of the form y = a * (1 + r)^t. Here's how to calculate it:
(a) Determine the equation for the number of bacteria, y, in x hours:
In this case, we know that the initial number of bacteria is 100, and the doubling period is 0.32 hours. The doubling period represents the amount of time it takes for the population to double in size.
To calculate the growth rate, we can use the formula: r = 1/doubling period.
So, in this case, r = 1/0.32 = 3.125.
Now we can substitute the values into the equation:
y = 100 * (1 + 3.125)^x.
(b) Determine how many hours will pass before there are 450 bacteria in the culture:
To find the time required for the number of bacteria to reach 450, we will substitute the value of y as 450 into the equation we derived in part (a) and solve for x.
450 = 100 * (1 + 3.125)^x.
Now, isolate the exponential term:
(1 + 3.125)^x = 450/100.
Taking the logarithm of both sides, we can solve for x:
log((1 + 3.125)^x) = log(450/100).
x * log(1 + 3.125) = log(4.5).
x = log(4.5) / log(1 + 3.125).
Using a calculator, we find:
x ≈ 3.102 hours.
Therefore, it will take approximately 3.102 hours for the culture to reach 450 bacteria.
a) y = 100 * 2^(x/0.32) (intuitively, after 0.32h, it doubles; after .64h, it quadruples, etc)
b) plug in y = 450, and solve for x
you should get x = 0.32*log(4.5)/log(2)
which is about 0.694h