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 `(d V(t))/(dt)`=-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 : (1) `T^2-1/k` (2) `I-(k T^2)/2` (3) `I-(k(T-t)^2)/2` (4) `e^(-k T)`

Options

A. `e^(-kT)`
B. `T^(2)-I/k`
C. `I-(kT^(2))/2`
D. `I-(k(T-t)^(2))/(2)`

Explanation

To solve the problem, we need to find the scrap value \( V(T) \) of the equipment after it has been used for \( T \) years, given the differential equation for depreciation. Let's go through the steps systematically.

### Step 1: Write down the differential equation
We are given the differential equation:
\[ \frac{dV(t)}{dt} = -k(T - t) \]
where \( k > 0 \) is a constant and \( T \) is the total life in years of the equipment.
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