The rate of bacteria growth in a laboratory experiment was at 16% per hour. If this experiment is repeated and
begins with 5 grams of bacteria, how much bacteria should be expected after 12 hours? Round to the nearest
tenth of a gram.
db/dt = .16 b
db/b = .16 dt
ln b = .16 t + c
e^ln b = b = e^(.16 t +c) = C e^.16 t
or more simply
b = C e^.16 t
when t =0 , e^.16 t = 1
so
C = value when t = 0
b = 5 e^(.16*12)
b = 34.1 grams
To calculate the amount of bacteria after 12 hours, we can use the formula for exponential growth:
A = P * (1 + r/100)^t
Where:
A = Final amount of bacteria
P = Initial amount of bacteria
r = Growth rate per hour
t = Time in hours
Given that the growth rate is 16% per hour, we can substitute the values into the formula:
P = 5 grams (Initial amount of bacteria)
r = 16% = 16/100 = 0.16 (Growth rate per hour)
t = 12 hours (Time in hours)
Now we can calculate the final amount of bacteria:
A = 5 * (1 + 0.16)^12
Using a calculator, we can simplify this expression:
A ≈ 5 * (1.16)^12
A ≈ 5 * 3.172364561
A ≈ 15.861822805
Rounding to the nearest tenth of a gram, the expected amount of bacteria after 12 hours is approximately 15.9 grams.