Question

The value of a car decreases at an annual rate of 9.9%. It is currently worth $15000. When will the car be worth $100?

Answer 1

The car will be worth $100 after 48 years and 23 days.

Explanation:

To decrease a number `x` by 9.9%, you must calculate

`x*(1-9.9/100) = x*0.901`

Be `x_0` the car's initial value, `x_1` its value after one year, `x_2` its value after two years, etc.

`x_1 = x_0*0.901`
`x_2=x_1*0.901=x_0*0.901*0.901=x_0*(0.901)^2`
`x_y=x_0*(0.901)^y` with `y` the number of years that passed.

Therefore, the car's value on year `y` is

`15000(0.901)^y`

You want to know when the value will drop to $100, so you must solve this equation:

`15000(0.901)^y=100`
`0.901^y=1/150`

Turn the power into a factor with the `log` function:
`color(gray)(log(1)=0;log(a/b)=log(a)-log(b);log(a^b)=blog(a))`

`log(0.901^y)=log(1/150)`
`ylog(0.901)=-log(150)`
`y=-log(150)/log(0.901) ~~ 48.064` years `~~48` years and `23` days