A car is worth $18000 and loses 15% of its value each year. Write… Step by Step

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The question

Question: A car is worth $18000 and loses 15% of its value each year.

Write the model.

1 Find the start

t is the time in years.

We call the value V.

V = value, t = years

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

Then the value is $18000.

That is the start.

start, t = 0: V = $18000

2 Find the factor

Each year, 15% of the value is lost.

What is left is 100% take away 15%: 85%.

100% − 15% = 85%

Percent means out of 100.

So 85% is 85/100: 0.85.

85% = 85100 = 0.85

So the factor is 0.85: each year, we multiply by 0.85.

factor: × 0.85 each year

3 Make a small table

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

Each step, we multiply by 0.85: 18000 times 0.85 is 15300.

Times 0.85 again is 13005.

After 2 years, we multiplied by 0.85 two times.

Two times is the power 2: 0.85 squared.

t = 2: 18000 × 0.85 × 0.85 = 18000 × 0.85²

After t years, we multiply t times.

So the power is t.

after t years: × 0.85, t times = × 0.85^t

4 Write the model

Start times the factor, to the power t.

So the model is V = 18000 times 0.85 to the power t.

V = 18000 × 0.85^t

Careful!

Not 0.15.

That is the part that is lost.

We multiply by the part that is left: 0.85.

Careful: not V = 18000 × 0.15^t ✗

5 Check

Let's check the first year the story way.

15% of 18000 is 0.15 times 18000: 2700.

15% of 18000 = 0.15 × 18000 = 2700

It loses 2700: 18000 minus 2700 is 15300.

18000 − 2700 = 15300

The model at t = 1 gives 18000 times 0.85: 15300 too.

It works!

model, t = 1: 18000 × 0.85¹ = 15300

Let's look back.

First: find the start.

Then: find the factor.

Then: make a small table.

Then: write the model.

Last: check.

Answer