A town has 4000 people and grows 2.5% each year. Write the model. Step by Step

Step by step

The question

Question: A town has 4000 people and grows 2.5% 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 4000 people.

That is the start.

start, t = 0: P = 4000

2 Find the factor

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

And 2.5% more is added.

So it becomes 102.5% of what it was.

100% + 2.5% = 102.5%

Percent means out of 100.

So 102.5% is 102.5/100: 1.025.

102.5% = 102.5100 = 1.025

So the factor is 1.025: each year, we multiply by 1.025.

factor: × 1.025 each year

3 Make a small table

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

Each step, we multiply by 1.025: 4000 times 1.025 is 4100.

Times 1.025 again is 4202.5.

After 2 years, we multiplied by 1.025 two times.

Two times is the power 2: 1.025 squared.

t = 2: 4000 × 1.025 × 1.025 = 4000 × 1.025²

After t years, we multiply t times.

So the power is t.

after t years: × 1.025, t times = × 1.025^t

4 Write the model

Start times the factor, to the power t.

So the model is P = 4000 times 1.025 to the power t.

P = 4000 × 1.025^t

Careful!

Not 0.025.

Times 0.025 would leave only a small part.

We keep all of it and add 2.5%: times 1.025.

Careful: not P = 4000 × 0.025^t ✗

5 Check

Let's check the first year the story way.

2.5% of 4000 is 0.025 times 4000: 100.

2.5% of 4000 = 0.025 × 4000 = 100

It gets 100: 4000 plus 100 is 4100.

4000 + 100 = 4100

The model at t = 1 gives 4000 times 1.025: 4100 too.

It works!

model, t = 1: 4000 × 1.025¹ = 4100

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer