Step by step
The question
Question: 500 bacteria double every 4 hours.
Write the model.
1 Find the start
t is the time in hours.
We call the number N.
At the start no time has passed: t = 0.
Then the number is 500.
That is the start.
2 Find the factor
To double means two times as much: times 2.
So the factor is 2: every 4 hours, we multiply by 2.
3 Make a small table
A small table: t = 0, 4, 8: one step is 4 hours.
Each step, we multiply by 2: 500 times 2 is 1000.
Times 2 again is 2000.
After two steps, 8 hours, we multiplied by 2 two times.
Two times is the power 2: 2 squared.
4 Count the steps
One step takes 4 hours.
At t = 4: 4/4 is 1 step.
At t = 8: 8/4 is 2 steps.
Like the table.
So in t hours there are t/4 steps.
We divide the time by 4, the length of one step.
That is the power.
5 Write the model
Start times the factor, to the power t/4.
So the model is N = 500 times 2 to the power t/4.
Careful!
Not 2 to the power 4 t.
That would be 4 steps every hour.
One step takes 4 hours, so we divide: t/4.
6 Check
Let's check one more step, t = 12.
12/4 is 3 steps.
2 to the power 3 is 8.
500 times 8 is 4000.
One more step of the table: 2000 times 2 is 4000 too.
It works!
Let's look back.
First: find the start.
Then: find the factor.
Then: make a small table.
Then: count the steps.
Then: write the model.
Last: check.