SOLUTION: The count in a bacterial culture was 400 after 2 hours and 25,600 after 6 hours. We are assuming exponential growth. What is the rate of growth of the population of bact

Question 135705: The count in a bacterial culture was 400 after 2 hours and 25,600 after 6 hours.
We are assuming exponential growth.

What is the rate of growth of the population of bacteria?
What was the initial population at time t = 0 hours?
Write the function that models the population n(t) after t hours.
When will the number of bacteria exceed 100,000?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the general exponential equation

Plug in and

Divide both sides by to isolate "a"

-----------------

Start with the general exponential equation

Plug in and

Plug in and

Multiply

Divide by subtracting the exponents

Divide both sides by 400

Take the fourth root of both sides

Simplify

-----------------------

Go back to the general exponential equation

Plug in , and

Square to get 8

Divide both sides by 8 to isolate "a"

So our equation is or if you want an approximation of then the equation is

a)
"What is the rate of growth of the population of bacteria?"

From the equation, the value of "b" is the rate of growth. So the rate of growth is or 2.82843

b) "What was the initial population at time t = 0 hours?"

Start with the equation we just found

Plug in

Raise to the zeroth power to get 1

Multiply

So the initial population is 50

c) "Write the function that models the population n(t) after t hours."

Earlier we found the equation to be or

d) "When will the number of bacteria exceed 100,000?"

Start with the equation we just found

Plug in

Divide both sides by 50

Rewrite as

Multiply the exponents

Take the log of both sides

Rewrite the right side

Divide both sides by

Use a calculator to evaluate the left side

Multiply both sides by 2

So it takes about 7.3 hours for the population to exceed 100,000
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