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 :

Options

A. ` I-(kT^(2))/2`
B. `I-k(T-10)^2/2`
C. `e^(-kT)`
D. `T^(2) -1/k`

Answer

a

Explanation

Given , ` (d{V(t)})/(dt) = -k (T-t)`
` :. d{V(t)} = - k (T-t) dt" "` …(i)
when t = 0, then V (t)=I
` int_(0)^(T)d {V(t)} = int_(0)^(T)- k(T-t)dt`
` rArr V (T) - V(0) = k [ (t-T)^(2)/2]_(0)^(T)`
` rArr V(T) -I = k/2 [{(T-T)^(2)-(0-T)^(2)]`
` :. V (T) = I - k/2 T^(2)`