Step by step
The question
Question: 80 mg of a medicine halves every 6 hours.
Write the model.
1 Find the start
t is the time in hours.
We call the amount A.
At the start no time has passed: t = 0.
Then the amount is 80 mg.
That is the start.
2 Find the factor
To halve means to cut in half: times one half.
So the factor is one half: every 6 hours, we multiply by one half.
3 Make a small table
A small table: t = 0, 6, 12: one step is 6 hours.
Each step, we multiply by one half: 80 times one half is 40.
Times one half again is 20.
After two steps, 12 hours, we multiplied by one half two times.
Two times is the power 2: one half squared.
4 Count the steps
One step takes 6 hours.
At t = 6: 6/6 is 1 step.
At t = 12: 12/6 is 2 steps.
Like the table.
So in t hours there are t/6 steps.
We divide the time by 6, the length of one step.
That is the power.
5 Write the model
Start times the factor, to the power t/6.
So the model is A = 80 times one half to the power t/6.
Careful!
Not one half to the power 6 t.
That would be 6 steps every hour.
One step takes 6 hours, so we divide: t/6.
6 Check
Let's check one more step, t = 18.
18/6 is 3 steps.
One half to the power 3 is 1/8.
80 times 1/8 is 10.
One more step of the table: 20 times one half is 10 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.