Step by step
The question
Question: 160 mg of a medicine halves every 8 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 160 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 8 hours, we multiply by one half.
3 Make a small table
A small table: t = 0, 8, 16: one step is 8 hours.
Each step, we multiply by one half: 160 times one half is 80.
Times one half again is 40.
After two steps, 16 hours, we multiplied by one half two times.
Two times is the power 2: one half squared.
4 Count the steps
One step takes 8 hours.
At t = 8: 8/8 is 1 step.
At t = 16: 16/8 is 2 steps.
Like the table.
So in t hours there are t/8 steps.
We divide the time by 8, the length of one step.
That is the power.
5 Write the model
Start times the factor, to the power t/8.
So the model is A = 160 times one half to the power t/8.
Careful!
Not one half to the power 8 t.
That would be 8 steps every hour.
One step takes 8 hours, so we divide: t/8.
6 Check
Let's check one more step, t = 24.
24/8 is 3 steps.
One half to the power 3 is 1/8.
160 times 1/8 is 20.
One more step of the table: 40 times one half is 20 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.