Question

Let $I$ be the purchase value of 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 the differential equation $\dfrac{{dV(t)}}{{dt}} = - k(T - t),$ where $k > 0$ is a constant and $T$ is the total life in years to the equipment. Then the scrap value $V(T)$ of the equipment is:

$${\text{(A) }}{{\text{T}}^2} - \dfrac{I}{k} \\\ {\text{(B) }}I - \dfrac{{k{T^2}}}{2} \\\ {\text{(C) }}I - \dfrac{{k{{(T - t)}^2}}}{2} \\\ {\text{(D) }}{e^{ - kT}} \\\$$

Answer 1

In the problem, it has been asked what is the scrap value of the equipment, i.e., what is the value of the equipment when$T = t$. To find out the total life of the equipment, we shall have to integrate the function $\dfrac{{dV(t)}}{{dt}} = - k(T - t),$with respect to $t$ ranging from $t = 0$ to $t = T$.

Complete step-by-step answer:
First, write the expression given in the question as below,
$\dfrac{{dV}}{{dt}} = - k(T - t)$
We know $\int x = \dfrac{x^2}{2}$ Here $x = k(T-t)$
So, integrating the above expression with respect to t ranging from $t = 0$ to $t = T$,

$$\dfrac{{dV}}{{dt}} = - k(T - t) \\\ \Rightarrow \int {dV} = - \int\limits_{}^{} {k(T - t)dt} \\\ \Rightarrow V = - \left[ { - \dfrac{{k{{(T - t)}^2}}}{2}} \right] + c{\text{ }}...........{\text{ (1)}} \\\$$

Now, we know that at $t = 0$,$V(t) = I$, therefore at $t = 0$,

$$\Rightarrow I = - \left[ { - \dfrac{{k{{(T)}^2}}}{2}} \right] + c \\\ \Rightarrow c = I - \dfrac{{k{{(T)}^2}}}{2} \\\$$

Putting this value of $c$in the equation (1), we will get,
$$$$$$\Rightarrow V = - \left[ { - \dfrac{{k{{(T - t)}^2}}}{2}} \right] + I - \dfrac{{k{{(T)}^2}}}{2}$$
To find out the scrap value of the equipment put$t = T$, we will get the following expression,

$$\Rightarrow {V_{Scrap}} = - \left[ { - \dfrac{{k{{(T - T)}^2}}}{2}} \right] + I - \dfrac{{k{{(T)}^2}}}{2} \\\ \Rightarrow {V_{Scrap}} = I - \dfrac{{k{{(T)}^2}}}{2} \\\$$

So, the correct answer is “Option B”.

Note: Generally, in these types of problems, integration of the function given in the question solves the problems at greater extent. You shall have to identify the range by carefully reading the question. The hint in the question was given that we shall have to find out the scrap value, it means the equipment has been completely utilised and that represents the total years of life, i.e.$T$.