A colony of bacteria doubles every 12 hours.
a. Write an exponential function that represents the population of the bacteria after t hours?
b. If the colony had 400 bacteria after 4 days, how many bacteria were there?
a. To write an exponential function that represents the population of the bacteria after t hours, we need to use the formula for exponential growth:
P(t) = P0 * (1+r)^t
Where:
- P(t) represents the population of the bacteria after t hours.
- P0 represents the initial population of the bacteria.
- r represents the growth rate (in decimal form).
- t represents the time in hours.
In this case, since the bacteria doubles every 12 hours, the growth rate r would be 1 (or 100%) plus the doubling factor, which is 2. Therefore, r = 2 - 1 = 1.
Let's write the exponential function for this scenario:
P(t) = P0 * (1+1)^t
P(t) = P0 * 2^t
b. The colony had 400 bacteria after 4 days, which is equal to 4 days * 24 hours/day = 96 hours.
Let's plug this value into the exponential function to find P0:
400 = P0 * 2^96
To solve for P0, divide both sides of the equation by 2^96:
P0 = 400 / 2^96
Calculating this value, we find:
P0 ≈ 4.764 * 10^28
Therefore, there were approximately 4.764 * 10^28 bacteria at the start.