A lake has 800 fish and the number grows 10% each year. Write the… Step by Step

Step by step

The question

Question: A lake has 800 fish and the number grows 10% each year.

Write the model.

1 Find the start

t is the time in years.

We call the number N.

N = number, t = years

At the start no time has passed: t = 0.

Then the number is 800.

That is the start.

start, t = 0: N = 800

2 Find the factor

Each year, all of the number stays: that is 100%.

And 10% more is added.

So it becomes 110% of what it was.

100% + 10% = 110%

Percent means out of 100.

So 110% is 110/100: 1.1.

110% = 110100 = 1.1

So the factor is 1.1: each year, we multiply by 1.1.

factor: × 1.1 each year

3 Make a small table

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

Each step, we multiply by 1.1: 800 times 1.1 is 880.

Times 1.1 again is 968.

After 2 years, we multiplied by 1.1 two times.

Two times is the power 2: 1.1 squared.

t = 2: 800 × 1.1 × 1.1 = 800 × 1.1²

After t years, we multiply t times.

So the power is t.

after t years: × 1.1, t times = × 1.1^t

4 Write the model

Start times the factor, to the power t.

So the model is N = 800 times 1.1 to the power t.

N = 800 × 1.1^t

Careful!

Not 0.1.

Times 0.1 would leave only a small part.

We keep all of it and add 10%: times 1.1.

Careful: not N = 800 × 0.1^t ✗

5 Check

Let's check the first year the story way.

10% of 800 is 0.1 times 800: 80.

10% of 800 = 0.1 × 800 = 80

It gets 80: 800 plus 80 is 880.

800 + 80 = 880

The model at t = 1 gives 800 times 1.1: 880 too.

It works!

model, t = 1: 800 × 1.1¹ = 880

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer