Question

Cholera Bacteria Suppose that a colony of bacteria starts with 1 bacterium and doubles in number every half hour. How many bacteria will the colony contain at the end of 24 \(\mathrm{h}\) ?

Short answer

The colony will contain \(2^{48}\) bacteria at the end of 24 hours.

Step-by-step solution

Understanding the Time Frame

Considering that the bacterium doubles in number every half hour, and that we need to find the number of bacteria at the end of 24 hours. Thus, the number of half hours in 24 hours is \(24 \times 2 = 48.\)

Applying the Geometric Progression Formula

The geometric progression formula for this situation can be written as \(P = P_0 \times 2^n\), where \(P\) is the final population, \(P_0\) is the initial population, and \(n\) is the number of time intervals. The initial population \(P_0\) is given as 1 bacterium. The factor 2 represents the doubling every half hour and \(n\) is the number of half-hour intervals in 24 hours, which we calculated in Step 1 as 48.

Substituting the values into the formula

Substituting these values into the formula, we have \(P = 1 \times 2^{48}\).

Calculating the final population

Evaluating this expression will give us the total number of bacteria at the end of 24 hours.