Question

Growth of Cholera Bacteria Suppose that the cholera bacteria in a colony grows unchecked according to the Law of Exponential Change. The colony starts 1 bacterium and doubles in number every half hour. (a) How many bacteria will the colony contain at the end of 24 h? (b) Writing to Learn Use part (a) to explain why a person who feels well in the morning may be dangerously ill by evening even though, in an infected person, many bacteria are destroyed.

Short answer

The colony will contain approximately \(2^{48}\) bacteria after 24 hours. This large number showcases the danger of bacterial infections and why quick medical response is important, as the bacteria multiply rapidly and can quickly overwhelm a person's immune system.

Step-by-step solution

Identify Given Variables

The given variables in this problem are \(N_0 = 1\) (the initial quantity of bacteria), \(t = 24*2\) (the time passed in half hours because the bacteria doubles every half-hour and we want to find the result after 24 hours), and \(k =1\) (as the bacteria doubles every half an hour).

Substitute Into The Formula

Substitute the given variables into the formula \(N = N_0 * 2^{t/k}\). So the equation will look like this: \(N = 1* 2^{24*2/1} = 2^{48}\).

Calculation

Calculate the value of \(2^{48}\), which equals a very large number, representing the amount of bacteria after 24 hours.

Explain the Growth

The reason a person can feel ill by the evening even though in the morning they felt fine is explained by the exponential growth of the bacteria. Although the body may initially be effective at combating the bacteria, the exponential rate of growth of the bacteria can eventually overcome the body's defenses.

Summarize

As a result, because of the rapid growth of bacteria, according to the exponential growth theory, the bacteria could grow to an enormous quantity in a short period of time which could lead to serious illness. This explains why even if many bacteria are destroyed, a person can still fall ill due to the extreme speed at which the bacteria multiply.