The solution for this problem involves the following steps:
First off, note the key given values:
The initial number of bacteria (n0) is 8800.
The number of bacteria after 1 hour (n) is 10000.
The time (t) which has passed is 1 hour.
Then, understand that an exponential growth model is used, which can be written as n = n0 * e^(rt), where n is the new number of bacteria, t is the time passed, r is the growth rate, and n0 is the initial number of bacteria.
Now, let's solve the questions given:
(a) Find a function n(t) that models the number of bacteria after t hours. (Round your r value to three decimal places.)
Firstly, we need to find the growth rate 'r'. We can find it by rearranging our growth model equation to 'r = ln(n/n0) / t'. Substituting the given values, we get:
r = ln(10000/8800) / 1
r ≈ 0.130
This implies that the growth rate is approximately 0.130.
Now we obtained r, we can write our function as:
n(t) = 8800 * e^(0.130t)
This function can now model the number of bacteria after t hours.
(b) Find the number of bacteria after 2 hours. (Round your answer to the nearest hundred)
To find number of bacteria after 2 hours, simply substitute t=2 into our function:
n(2) = 8800 * e^(0.130*2)
n(2) ≈ 10578.391
Rounding to the nearest hundred, we get:
n(2) ≈ 10600
Therefore, after 2 hours, the number of bacteria will be approximately 10600.
(c) After how many hours will the number of bacteria double?
We can solve this by using the formula 't = ln(2) / r' which represents the doubling time form our exponential growth model.
Substituting the calculated 'r' into our formula we get:
t = ln(2) / 0.130
t ≈ 5.33
Therefore, the number of bacteria will double after approximately 5.33 hours.