The value V of a factory machine depreciated with time t years such that dV/dt=-kV, for some constant k>0.
(i) Show that V=V_0 e^(-kt) satisfies the given differential equation.
(ii) The initial value of a particular item of machinery is $15000. Explain why V_0=15000.
(iii) In the first year the machine depreciates in value by 30%. Find the value of the constant k.
(iv) The company that bought the machine writes off any machine when it has depreciated to 5% of its initial value. How many years does this take? Round your answer to the nearest whole year.
I am not good with starting growth and decay questions. Im just trying to do this problem. Any solutions or starting points to quickly do this with working would be appreciated.
(i) $\frac{dv}{dt} = -kV$
Separate variables
$\frac{dV}{V} = -k\cdot dt$
$\ln V = -kt + C$
$V = e^C\cdot e^{-kt}$
When $t = 0$ then....
$V_0 = e^C$
So $V = V_0\cdot e^{-kt}$
(ii) $V_0$ is the value when $t = 0$ which is $\$15 000$ the initial value .
(iii) When $t = 1$ then....
$10500 = 15000\cdot e^{-k}$
$\frac{10500}{15000} = e^{-k}$
$\ln(0.7) = -k$
$k = 0.356675$
(iv) $750 = 15000e^{-0.356675t}$
$.05 = e^{-0.356675t}$
$\ln(.05) = -0.356675t$
$t = \frac{-2.995732}{-0.356675}$
$t = 8$ years