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 equationd 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

$straight T squared minus 1 over straight k$
$straight I space minus KT squared over 2$
$straight I space minus space kT squared over 2$

e^-kT

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Solution

straight I space minus KT squared over 2

fraction numerator dv left parenthesis straight t right parenthesis over denominator dt end fraction space equals space straight k space left parenthesis straight T minus straight t right parenthesis integral dv left parenthesis straight t right parenthesis space equals space integral left parenthesis negative kT right parenthesis dt space plus integral ktdt straight V left parenthesis straight t right parenthesis space equals space minus space kTt space plus space straight k straight t squared over 2 space plus straight C at space straight t space equals space 0 space space straight C space equals space straight I straight V left parenthesis straight T right parenthesis space equals space minus space kTt space plus space kT squared over 2 plus straight I Now space at space straight t space equals straight T straight V left parenthesis straight T right parenthesis space equals space minus space kT squared space plus straight k space straight T squared over 2 plus straight I straight V space left parenthesis straight T right parenthesis space equals space straight I minus 1 half kT squared

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