Question

a certain species of bacteria doubles in number every 3 hours. if you start with 500 bacteria, how many bacteria will there be after 12 hours? is this an example of exponential growth or decay? explain your reasoning and describe how changing the doubling time would affect the population size.

Explanation:

Step1: Determine the number of doubling periods

The bacteria doubles every 3 hours, and we want to find the number after 12 hours. So the number of doubling periods \( n=\frac{12}{3} = 4\).

Step2: Calculate the final number of bacteria

The formula for exponential growth (when doubling) is \( N = N_0\times2^n\), where \( N_0 = 500\) (initial number), \( n = 4\) (number of doubling periods). So \( N=500\times2^4\).
\( 2^4=16\), then \( N = 500\times16=8000\).

Step3: Determine if it's growth or decay

Since the number of bacteria is increasing over time (doubling), this is exponential growth. Exponential growth occurs when the quantity increases by a constant factor over equal time intervals (here, the factor is 2 every 3 hours).

Step4: Effect of changing doubling time

If the doubling time decreases (e.g., doubles every 2 hours instead of 3), the number of doubling periods in a given time (like 12 hours) increases, so the population will grow faster. If the doubling time increases (e.g., doubles every 4 hours), the number of doubling periods in the same time decreases, so the population will grow slower.

Answer:

After 12 hours, there will be 8000 bacteria. This is an example of exponential growth because the bacteria population is multiplying by a constant factor (2) over equal time intervals (3 hours). A shorter doubling time would make the population grow faster (more doubling periods in the same total time), and a longer doubling time would make the population grow slower (fewer doubling periods in the same total time).