If a broth is inoculated with 4 bacteria and has a generation time of 20 minutes, how many bacteria will there be after 2 hours?

To determine the number of bacteria after a specific time period, we can use the formula for exponential growth, which is:

[ N = N_0 \times 2^{(t/g)} ]

In this formula:

• ( N ) is the final number of bacteria.
• ( N_0 ) is the initial number of bacteria.
• ( t ) is the total time of growth (in the same units as the generation time).
• ( g ) is the generation time.

In the given scenario:

• The initial number of bacteria ( N_0 ) is 4.
• The generation time ( g ) is 20 minutes.
• The total time ( t ) is 2 hours, which is equal to 120 minutes.

To find how many generations fit into the 2-hour period, we calculate:

[ \frac{t}{g} = \frac{120 \text{ minutes}}{20 \text{ minutes}} = 6 ]

Now, we can substitute these values into the exponential growth formula:

[ N = 4 \times 2^6 ]

Next, we calculate ( 2^6 ):

[ 2^6 = 64 ]