SOLUTION: The count in a culture of bacteria was 200 after 2 hours and 25000 after 5 hours. a. Find a function that models the number of bacteria n(t) after t hours. b. When does the cultu

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Question 1101289: The count in a culture of bacteria was 200 after 2 hours and 25000 after 5 hours.
a. Find a function that models the number of bacteria n(t) after t hours.
b. When does the culture contain 2,000 bacteria?
I am taking an online pre-calc math class. Our math teacher does not make videos for us but I have a text book. However, I feel like the text book does not have enough examples for us and I need help on this problem. I would appreciate to see the work so I can understand the problem. Thank you!
Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

The number of bacteria is increasing exponentially, so the function is of the form

where a is an initial value (when x=0) and b is the growth rate.

(1) Use the two given pieces of information to write equations in that form, using x as the time in hours:

(2) Divide the second equation by the first; that will eliminate a, allowing you to find the value of b:

Remember not to just go through the motions of solving the problem; take the time to realize that b=5 means the number of bacteria grows by a factor of 5 each hour.
(You will appreciate the mathematics more if you understand what it is doing....)

(3) Use the value you have found for b in either of the original equations to find the value of a:

So the function is

For part b, you need to solve the equation

Up to now, the numbers have worked out nicely; a and b are whole numbers. That will no longer be the case:
f(0) = 8
f(1) = 40
f(2) = 200
f(3) = 1000
f(4) = 5000
f(5) = 25000
So the solution to part b will be between 3 and 4 hours.

To solve algebraically, because the variable is in an exponent, you need to use logarithms.

(to 2 decimal places)