A city has 60000 people and grows 4% each year. Write the model. Step by Step

Step by step

The question

Question: A city has 60000 people and grows 4% 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 60000 people.

That is the start.

start, t = 0: P = 60000

2 Find the factor

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

And 4% more is added.

So it becomes 104% of what it was.

100% + 4% = 104%

Percent means out of 100.

So 104% is 104/100: 1.04.

104% = 104100 = 1.04

So the factor is 1.04: each year, we multiply by 1.04.

factor: × 1.04 each year

3 Make a small table

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

Each step, we multiply by 1.04: 60000 times 1.04 is 62400.

Times 1.04 again is 64896.

After 2 years, we multiplied by 1.04 two times.

Two times is the power 2: 1.04 squared.

t = 2: 60000 × 1.04 × 1.04 = 60000 × 1.04²

After t years, we multiply t times.

So the power is t.

after t years: × 1.04, t times = × 1.04^t

4 Write the model

Start times the factor, to the power t.

So the model is P = 60000 times 1.04 to the power t.

P = 60000 × 1.04^t

Careful!

Not 0.04.

Times 0.04 would leave only a small part.

We keep all of it and add 4%: times 1.04.

Careful: not P = 60000 × 0.04^t ✗

5 Check

Let's check the first year the story way.

4% of 60000 is 0.04 times 60000: 2400.

4% of 60000 = 0.04 × 60000 = 2400

It gets 2400: 60000 plus 2400 is 62400.

60000 + 2400 = 62400

The model at t = 1 gives 60000 times 1.04: 62400 too.

It works!

model, t = 1: 60000 × 1.04¹ = 62400

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer