If a one-liter jar holds \(10^{16}\) bacteria, how many bacteria would we start in the jar so that the jar reaches full capacity after 24 hours if we increase the doubling time to a more modest/realistic 30 minutes?

As the doubling time is 30 minutes and we have 24 hours before the jar reaches its full capacity, we first need to calculate the total number of doubling intervals in the given timeframe. Since there are 60 minutes in an hour and we have 24 hours, there are (24 * 60) minutes in total. Divide this by the doubling time to find the number of intervals: Number of doubling intervals = (24 * 60) / 30

Now, we will use the exponential growth formula to find the initial number of bacteria. The exponential growth formula is: Final amount = Initial amount * 2^(number of doubling intervals) We are given the final amount as \(10^{16}\) bacteria, and we need to find the initial amount. Rearrange the formula to solve for the initial amount: Initial amount = Final amount / 2^(number of doubling intervals) Substitute the given values into the formula and calculate the initial amount of bacteria: Initial amount = \(10^{16}\) / 2^((24 * 60) / 30) The initial number of bacteria that should be present in the jar to reach full capacity after 24 hours with a doubling time of 30 minutes is the calculated value of the initial amount.