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 $\frac{d V(t)}{d t}=-k(T-t),$ 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

• A. $e^{-k T}$
• B. $T^{2}-\frac{1}{k}$
• C. $I-\frac{k T^{2}}{2}$
• D. $I-\frac{k(T-t)^{2}}{2}$

Answer 1

Answer: (C)

$I-\frac{k T^{2}}{2}$

Answer 2

$\int\limits_{l}^{V(T)} d V(t)=\int_{t=0}^{T}-k(T-t) d t$ $\Rightarrow \quad V(T)-I=k\left[\frac{(T-t)^{2}}{2}\right]_{0}^{T}$ $\Rightarrow \quad V(T)-I=-k\left[\frac{T^{2}}{2}\right]$ $\Rightarrow \quad V(T)=I-\frac{k T^{2}}{2}$