Bacteria is growing at a rate of 15 percent per minute in a closed container. If the initial number of bacteria is two and the carrying capacity of the container is two million cells, how long will it take the bacteria to reach one million cells? Give your answer to the nearest minute.
Here, we have a question about population growth, but the population can’t grow to an unlimited level. We’re told that it has a carrying capacity of two million cells. That is the maximum population of bacteria that can be supported. We’re therefore going to use the logistic model for population growth to answer this question. And we can quote its standard solution.
According to the logistic model, a population with a growth rate of 𝐾, a carrying capacity of 𝐿, and an initial population of 𝑃 nought can be modelled as 𝑃 of 𝑡 is equal to 𝐿 over one plus 𝐴𝑒 to the power of negative 𝐾𝑡, where 𝐴 is equal to 𝐿 minus 𝑃 nought over 𝑃 nought. We’re given some of these values in the question.
Firstly, we’re told that bacteria is growing at a rate of 15 percent per minute. So, the growth rate 𝐾 is equal to 0.15. We’re also told that the initial number of bacteria is two, so that’s our value for 𝑃 nought. And we were told in the question that the carrying capacity of this container is two million, so that’s our value for 𝐿.
We can also work out the value of 𝐴 from the values of 𝐿 and 𝑃 nought. It’s two million minus two over two, which is 999999. So, substituting each of these values into the general solution, we know that the population of bacteria at time 𝑡 is equal to two million over 1 plus 999999𝑒 to the power of negative 0.15𝑡.
Now, let’s look at what we’re asked to do in the question. We’re asked for how long it will take for the bacteria to reach one million cells. That means we’re looking for the value of 𝑡 when 𝑃 is equal to one million. So, what we can do is substitute one million for 𝑃 and then solve the resulting equation to find 𝑡. So, doing so gives one million is equal to two million over 1 plus 999999𝑒 to the negative 0.15𝑡.
Before we go any further, we can simplify by dividing each side by one million to give just one on the left-hand side and two in the numerator on the right-hand side. We can then multiply by the denominator of the quotient to give one plus 999999𝑒 to the negative 0.15𝑡 is equal to two. Subtracting one from each side and then dividing by 999999 gives 𝑒 to the power of negative 0.15𝑡 is equal to one over 999999.
The next step is to take the natural logarithm of each side of the equation, knowing that this will cancel out the exponential on the left-hand side. Leaving us with negative 0.15𝑡 is equal to the natural logarithm of one over 999999. Using laws of logarithms, we can express the right-hand side as negative the natural logarithm of 999999, if we wish. And then, we see that by multiplying or dividing both sides of the equation by negative one, we can eliminate those negative signs.
The final step in solving for 𝑡 is to divide both sides of the equation by 0.15. We can then use our calculators to evaluate this, and it gives 92.1033 continuing. Looking back at the question, we see that we’re asked to give our answer to the nearest minute. So, round into the nearest integer value, we find that it will take 92 minutes for the bacteria to reach one million cells.