Step by step
The question
Question: 200 cells double every 3 days.
Write the model.
1 Find the start
t is the time in days.
We call the number N.
At the start no time has passed: t = 0.
Then the number is 200.
That is the start.
2 Find the factor
To double means two times as much: times 2.
So the factor is 2: every 3 days, we multiply by 2.
3 Make a small table
A small table: t = 0, 3, 6: one step is 3 days.
Each step, we multiply by 2: 200 times 2 is 400.
Times 2 again is 800.
After two steps, 6 days, we multiplied by 2 two times.
Two times is the power 2: 2 squared.
4 Count the steps
One step takes 3 days.
At t = 3: 3/3 is 1 step.
At t = 6: 6/3 is 2 steps.
Like the table.
So in t days there are t/3 steps.
We divide the time by 3, the length of one step.
That is the power.
5 Write the model
Start times the factor, to the power t/3.
So the model is N = 200 times 2 to the power t/3.
Careful!
Not 2 to the power 3 t.
That would be 3 steps every day.
One step takes 3 days, so we divide: t/3.
6 Check
Let's check one more step, t = 9.
9/3 is 3 steps.
2 to the power 3 is 8.
200 times 8 is 1600.
One more step of the table: 800 times 2 is 1600 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.