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QJEEM11Ma97de: JEE Main 2011 Mathematics - Class 12 JEEM11Ma97de Solution: Let I be the purchase value of an equipment and V ( t ) be the…

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

Step-by-Step Verified Solution

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

\Rightarrow \int d V t = - k \int (T - t) d t

V (t) = \frac{k {(T - t)}^{2}}{2} + c

V (0) = I \Rightarrow I = \frac{K T^{2}}{2} + C

\Rightarrow C = I - \frac{K T^{2}}{2}

∴

V (T) = 0 + C = I - \frac{K T^{2}}{2}