A town has 5000 people and grows 2% each year. Write the model. Step by Step

Step by step

The question

Question: A town has 5000 people and grows 2% 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 5000 people.

That is the start.

start, t = 0: P = 5000

2 Find the factor

Each year, all the people stay: that is 100%.

And 2% more is added.

So it becomes 102% of what it was.

100% + 2% = 102%

Percent means out of 100.

So 102% is 102/100: 1.02.

102% = 102100 = 1.02

So the factor is 1.02: each year, we multiply by 1.02.

factor: × 1.02 each year

3 Make a small table

A small table: t = 0, 1, 2.

Each step, we multiply by 1.02: 5000 times 1.02 is 5100.

Times 1.02 again is 5202.

After 2 years, we multiplied by 1.02 two times.

Two times is the power 2: 1.02 squared.

t = 2: 5000 × 1.02 × 1.02 = 5000 × 1.02²

After t years, we multiply t times.

So the power is t.

after t years: × 1.02, t times = × 1.02^t

4 Write the model

Start times the factor, to the power t.

So the model is P = 5000 times 1.02 to the power t.

P = 5000 × 1.02^t

Careful!

Not 0.02.

Times 0.02 would leave only a small part.

We keep all of it and add 2%: times 1.02.

Careful: not P = 5000 × 0.02^t ✗

5 Check

Let's check the first year the story way.

2% of 5000 is 0.02 times 5000: 100.

2% of 5000 = 0.02 × 5000 = 100

It gets 100: 5000 plus 100 is 5100.

5000 + 100 = 5100

The model at t = 1 gives 5000 times 1.02: 5100 too.

It works!

model, t = 1: 5000 × 1.02¹ = 5100

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer