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?

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Answer 1 · 2015-06-14T11:47:11

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

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