A forest has 3000 trees and loses 4% every 5 years. Write the model. Step by Step

Step by step

The question

Question: A forest has 3000 trees and loses 4% every 5 years.

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 3000.

That is the start.

start, t = 0: N = 3000

2 Find the factor

Every 5 years, 4% of the number is lost.

What is left is 100% take away 4%: 96%.

100% − 4% = 96%

Percent means out of 100.

So 96% is 96/100: 0.96.

96% = 96100 = 0.96

So the factor is 0.96: every 5 years, we multiply by 0.96.

factor: × 0.96 every 5 years

3 Make a small table

A small table: t = 0, 5, 10: one step is 5 years.

Each step, we multiply by 0.96: 3000 times 0.96 is 2880.

Times 0.96 again is 2764.8.

After two steps, 10 years, we multiplied by 0.96 two times.

Two times is the power 2: 0.96 squared.

t = 10: 3000 × 0.96 × 0.96 = 3000 × 0.96²

4 Count the steps

One step takes 5 years.

At t = 5: 5/5 is 1 step.

t = 5: 55 = 1 step

At t = 10: 10/5 is 2 steps.

Like the table.

t = 10: 105 = 2 steps

So in t years there are t/5 steps.

We divide the time by 5, the length of one step.

That is the power.

t: t5 steps → × 0.96^t5

5 Write the model

Start times the factor, to the power t/5.

So the model is N = 3000 times 0.96 to the power t/5.

N = 3000 × 0.96^t5

Careful!

Not 0.04.

That is the part that is lost.

We multiply by the part that is left: 0.96.

Careful: not N = 3000 × 0.04^t5 ✗

6 Check

Let's check the first 5 years the story way.

4% of 3000 is 0.04 times 3000: 120.

4% of 3000 = 0.04 × 3000 = 120

It loses 120: 3000 minus 120 is 2880.

3000 − 120 = 2880

The model at t = 5 gives 3000 times 0.96: 2880 too.

It works!

model, t = 5: 3000 × 0.96⁵⁵ = 2880

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: count the steps.

Then: write the model.

Last: check.

Answer