Let I be the purchase value of an equipment and V(t) be the value after it has been used f…

Question

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by differential equation $${{dv\left( t \right)} \over {dt}} =  - k\left( {T - t} \right),$$  where $k>0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is

Accepted Answer

$${{dV\left( t \right)} \over {dt}} =  - k\left( {T - t} \right)$$

$$ \Rightarrow \int {dVt}  =  - k\int {\left( {T - t} \right)} dt$$

$$V\left( t \right) = {{k{{\left( {T - t} \right)}^2}} \over 2} + c$$

$$V\left( 0 \right) = I \Rightarrow I = {{K{T^2}} \over 2} + C$$

$$ \Rightarrow C = I - {{K{T^2}} \over 2}$$

$$\therefore$$ $$\,\,\,\,\,V\left( T \right) = 0 + C = I - {{K{T^2}} \over 2}$$

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