Worked examples
The question
Question: A town has 2000 people and grows 3% each year.
Write the model.
1 Find the start
t is the time in years.
We call the number of people P.
P = people, t = years
At the start no time has passed: t = 0.
Then there are 2000 people.
That is the start.
start, t = 0: P = 2000
2 Find the factor
Each year, all the people stay: that is 100%.
And 3% more is added.
So it becomes 103% of what it was.
100% + 3% = 103%
Percent means out of 100.
So 103% is 103/100: 1.03.
103% = 103100 = 1.03
So the factor is 1.03: each year, we multiply by 1.03.
factor: × 1.03 each year
3 Make a small table
A small table: t = 0, 1, 2.
Each step, we multiply by 1.03: 2000 times 1.03 is 2060.
Times 1.03 again is 2121.8.
After 2 years, we multiplied by 1.03 two times.
Two times is the power 2: 1.03 squared.
t = 2: 2000 × 1.03 × 1.03 = 2000 × 1.032
After t years, we multiply t times.
So the power is t.
after t years: × 1.03, t times = × 1.03t
4 Write the model
Start times the factor, to the power t.
So the model is P = 2000 times 1.03 to the power t.
P = 2000 × 1.03t
Careful!
Not 0.03.
Times 0.03 would leave only a small part.
We keep all of it and add 3%: times 1.03.
Careful: not P = 2000 × 0.03t ✗
5 Check
Let's check the first year the story way.
3% of 2000 is 0.03 times 2000: 60.
3% of 2000 = 0.03 × 2000 = 60
It gets 60: 2000 plus 60 is 2060.
2000 + 60 = 2060
The model at t = 1 gives 2000 times 1.03: 2060 too.
It works!
model, t = 1: 2000 × 1.031 = 2060
Let's look back.
First: find the start.
Then: find the factor.
Then: make a small table.
Then: write the model.
Last: check.
Answer Careful: not P = 2000 × 0.03t ✗
▶ Watch it solved
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.
N = number, t = hours
At the start no time has passed: t = 0.
Then the number is 500.
That is the start.
start, t = 0: N = 500
2 Find the factor
To double means two times as much: times 2.
double = × 2
So the factor is 2: every 4 hours, we multiply by 2.
factor: × 2 every 4 hours
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.
t = 8: 500 × 2 × 2 = 500 × 22
4 Count the steps
One step takes 4 hours.
At t = 4: 4/4 is 1 step.
t = 4: 44 = 1 step
At t = 8: 8/4 is 2 steps.
Like the table.
t = 8: 84 = 2 steps
So in t hours there are t/4 steps.
We divide the time by 4, the length of one step.
That is the power.
t: t4 steps → × 2t4
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.
N = 500 × 2t4
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.
Careful: not N = 500 × 24t ✗
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.
model, t = 12: 500 × 2124 = 500 × 23 = 500 × 8 = 4000
One more step of the table: 2000 times 2 is 4000 too.
It works!
table, one more step: 2000 × 2 = 4000
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.
Answer Careful: not N = 500 × 24t ✗
▶ Watch it solved
The question
Question: $1000 grows 5% every 2 years.
Write the model.
1 Find the start
t is the time in years.
We call the amount A.
A = amount, t = years
At the start no time has passed: t = 0.
Then the amount is $1000.
That is the start.
start, t = 0: A = $1000
2 Find the factor
Every 2 years, all of the amount stays: that is 100%.
And 5% more is added.
So it becomes 105% of what it was.
100% + 5% = 105%
Percent means out of 100.
So 105% is 105/100: 1.05.
105% = 105100 = 1.05
So the factor is 1.05: every 2 years, we multiply by 1.05.
factor: × 1.05 every 2 years
3 Make a small table
A small table: t = 0, 2, 4: one step is 2 years.
Each step, we multiply by 1.05: 1000 times 1.05 is 1050.
Times 1.05 again is 1102.5.
After two steps, 4 years, we multiplied by 1.05 two times.
Two times is the power 2: 1.05 squared.
t = 4: 1000 × 1.05 × 1.05 = 1000 × 1.052
4 Count the steps
One step takes 2 years.
At t = 2: 2/2 is 1 step.
t = 2: 22 = 1 step
At t = 4: 4/2 is 2 steps.
Like the table.
t = 4: 42 = 2 steps
So in t years there are t/2 steps.
We divide the time by 2, the length of one step.
That is the power.
t: t2 steps → × 1.05t2
5 Write the model
Start times the factor, to the power t/2.
So the model is A = 1000 times 1.05 to the power t/2.
A = 1000 × 1.05t2
Careful!
Not 0.05.
Times 0.05 would leave only a small part.
We keep all of it and add 5%: times 1.05.
Careful: not A = 1000 × 0.05t2 ✗
6 Check
Let's check the first 2 years the story way.
5% of 1000 is 0.05 times 1000: 50.
5% of 1000 = 0.05 × 1000 = 50
It gets 50: 1000 plus 50 is 1050.
1000 + 50 = 1050
The model at t = 2 gives 1000 times 1.05: 1050 too.
It works!
model, t = 2: 1000 × 1.0522 = 1050
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.
Answer Careful: not A = 1000 × 0.05t2 ✗
▶ Watch it solved
