During a research​ experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 8 ​:00 AM there were 6 comma 0

Question

During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 8:00  AM there were 6,000 bacteria present in the culture. At noon, the number of bacteria grew to 6,700 How many bacteria will there be at midnight?

Answer 1

Function of exponential growth or decay:
f(t) = C*exp(kt)

f(0) = 6000 ---> C = 6000

f(4) = 6700 ----> 6700 = 6000*exp(4k)

67/60 = exp(4k)

ln 67 - ln 60 = 4k

ln 67 - ln 60
-------------- = k
4

k = ln(67/60)/4

 

So the function is
f(t) = 6000 * exp ( ln(67/60)/4 * t)


At midight, t = 16, becuase noon was t=4 and midnight is 12 hours later.

f(16) = 6000 * exp ( ln(67/60)/4*16) =
6000 * exp ( ln(67/60)*4) = <--- 16 cancels 4 in the denominator of the exponent
= 9329.2223....


Another way to view the same problem:
grew by 700 in 4 hours when population was 6000 --> 700/6000 = 11 2/3 % growth

11 2/3 % increase of 6700 --> 7481 by 4 pm.
11 2/3 % increase of 7481 --> 8353 by 8 pm
11 2/3 % increase of 8353 --> 9328 by midnight