Answer:
- p = 6912 after 15 hours
- p = 4000·e^(0.036463t)
Explanation:
We want to model the growth of a population of bacteria using an exponential function. We find it convenient to use an "exact" exponential function that relies directly on the numbers given in the problem statement. That function has the form ...
p(t) = (initial population) × (growth factor)^(t/(growth period))
where the "growth factor" is the multiplying factor of the population in period "growth period."
__
Exponential Function (1)
Here, the "growth factor" is 4800/4000 = 1.2, and the corresponding "growth period" is 5 hours. Then the population function can be ...
p(t) = 4000×1.2^(t/5)
__
Population
The population after 15 hours is ...
p(15) = 4000×1.2^(15/3) = 4000×1.728
p(15) = 6912
__
Exponential Function (2)
To change this to the form ae^(kt), we find the value of k to be ...
k = ln(1.2^(1/5)) = ln(1.2)/5 ≈ 0.036463
Then the desired form of the function for p is ...
p(t) = 4000·e^(0.036463t)
_____
Additional comment
In the form p=ae^(kt) the value of k is often rounded to 3 or 4 decimal places. We believe that best accuracy is obtained if k has at least the number of significant digits expected in the answer. Here, the population of almost 7000 has 4 significant digits, so we choose to express k to 5 significant figures. That still gives a small error relative to the first formula that is only "corrected" by rounding the result to an integer.