A bacteria that grows according to the function

yakige8425

yakige8425

Answered question

2022-08-30

A bacteria that grows according to the function ๐‘“(๐‘ก) = 500๐‘’^0.05๐‘ก
, where ๐‘ก is measured in minutes. 
a) How many bacteria are present in the population after 4 hours?
b) When does the population reach 100 million bacteria?

Answer & Explanation

nick1337

nick1337

Expert2023-05-29Added 777 answers

To solve the given problem, let's analyze the growth of bacteria according to the function:
f(t)=500e0.05t
where t is measured in minutes.
a) To find the number of bacteria present in the population after 4 hours, we need to convert the time to minutes since the function is given in terms of minutes.
1 hour = 60 minutes
4 hours = 4 * 60 = 240 minutes
Substituting t=240 into the function, we have:
f(240)=500e0.05ยท240
Simplifying the exponent, we get:
f(240)=500e12
Calculating this value, we find:
f(240)โ‰ˆ500ยท162754.7914โ‰ˆ81377395.7
Therefore, there are approximately 81,377,395 bacteria present in the population after 4 hours.
b) To find when the population reaches 100 million bacteria, we need to solve the equation:
f(t)=100ร—106
Substituting this into the given function, we have:
500e0.05t=100ร—106
Dividing both sides by 500, we get:
e0.05t=200000
To solve for t, we need to take the natural logarithm (ln) of both sides:
ln(e0.05t)=ln(200000)
Using the property of logarithms, we can bring down the exponent:
0.05tln(e)=ln(200000)
Since ln(e)=1, the equation simplifies to:
0.05t=ln(200000)
Now, we can solve for t by dividing both sides by 0.05:
t=ln(200000)0.05
Using a calculator, we find:
tโ‰ˆ12.2060.05โ‰ˆ244.12
Therefore, the population reaches 100 million bacteria approximately at 244.12 minutes.

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