Question

HELP ME IM GONNA CRY PLEASE

A car was purchased for $20,000. The car depreciates by 22% of each year.
a) What is the value of the car when it is 12 years
old?
b) How long will it take for the car to be worth less than $100?

Answer 1

Hello, a car was purchased for $20,000.

This is the initial value.

The car depreciates by 22% of each year.

After 1 year, the value is the initial value 20,000 minus 22% of 20,000.

`20000-20\%*20000=20000\cdot (1-20\%)=20000\cdot (1-0.20)=20000\cdot 0.8=16000`

After 2 years, the value is.

`20000\cdot 0.8\cdot 0.8=20000\cdot0.8^2=12800`

Let's take n a positive integer, after n years, the value is.

`\large \boxed{\sf \bf \ 20000\cdot0.8^n \ }`

a) After 12 years, the value is.

`20000\cdot0.8^(12)=1374.389...`

This is rounded to $1,374

b) We need to find n such that

`20000\cdot0.8^n=100\\\\ln(20000)+nln(0.8)=ln(100)\\\\n=(ln(100)-ln(20000))/(ln(0.8))=23.74...`

This is around 23.75 meaning 23 years and 75% of 1 year (meaning 9 months).

So to be worth less than $100, 23 years and 9 months are required.

Thank you