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?

1 Answer
Jun 14, 2015

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