Answer:
(a) y = 40·e^(1/6·ln(3.1)t)
(b) 1439
Step-by-step explanation:
You want the exponential formula that describes the growth of a population from 40 bacteria to 124 in 6 days, and you want the population after 19 days.
(a) Exponential formula
The exponential formula can be written as ...
y = (initial number) · (growth factor)^(t/(growth period))
Here, we have ...
- initial number = 40
- growth factor = 124/40 = 3.1
- growth period = 6 (days)
So, a suitable formula would be ...
[tex]y=40\cdot3.1^{\frac{t}{6}}\\\\y=40\cdot(3.1^{\frac{1}{6}})^t[/tex]
However, we don't want 3.1 to be the base of the exponential. Instead, we want it to be e. We know that ...
[tex]x=e^{\ln(x)}[/tex]
In this case, "x" is the value 3.1^(1/6), so the value multiplying t will be found from ...
[tex](3.1^\frac{1}{6})^t=e^{\ln(3.1^{1/6})t}=e^{\frac{1}{6}\ln(3.1)t}[/tex]
Then the full exponential equation is ...
[tex]\displaystyle \boxed{y=40\cdot e^{\frac{1}{6}\ln(3.1)t}}[/tex]
19 Days
Using t=19 in the above formula, we have ...
[tex]y=40\cdot e^{\frac{19}{6}\ln(3.1)}\approx 40\cdot e^{3.582773}\approx 1438.93[/tex]
After 19 days, there will be about 1439 bacteria.
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Additional comment
Our calculator maintains 32 significant digits. It is both cumbersome and unnecessary to show the full precision intermediate results in the answer here. For the purpose of this answer, the values shown are rounded to a precision sufficient to support the correct final answer. The actual calculations were done with full calculator precision.
