Question

The rate at which an item depreciates is proportional to its value at that instant. If an item is presently values at $70,000 and 10 months later it is valued at $22,222, when in years will it be valued at $19,999?

I'm not sure what formula to use...

Answer 1

Ah hah, a real calculus problem :)

dV/dt = - k V

dV/V = -k dt

ln V = -k t + C

e^ln V = V = e^(-kt+C) = c e^-kt

so here at t = 0
e^0 = 1
so V = c = 70,000
and
V = 70,000 e^-kt
now when t = 10/12 years, V = 22,222

22,222 = 70,000 e^-k(.83333)
ln (22,222/70,000) = - .83333 k
-1.1474= - .83333 k
k = 1.377
so
V = 70,000 e^-1.377 t
19,999 = 70,000 e^-1.377 t
ln(19,999/70,000) = -1.377 t
t = .9098 years

By the way, are you sure it was 10 months and not ten yars?

Answer 2

thank you and yes it was months