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Let I denote the buying price of a machine, and let V(t) represent its worth after t years of use. Suppose the depreciation follows the differential equation dV(t)/dt = -k(T - t), where k > 0 is a constant and T is the machine’s total service life in years. What is the scrap value V(T) of the machine?
- I - kT²/2
- I - k(T - t)²/2
- e^(-kT)
- T² - 1/k
Correct answer: I - kT²/2
Solution
The correct option represents the total depreciation of the machine over its entire service life, calculated by integrating the given differential equation. This results in the worth of the machine at the end of its life, which is the initial price minus the accumulated depreciation, leading to the expression I - kT²/2.
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