A town has 2000 people and grows 3% each year. Write the model. Step by Step

Step by step

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 = × 1.03^t

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 × 1.03^t

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 × 0.03^t ✗

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