GATE 2016 CS – Question 10
In a process, the number of cycles to failure decreases exponentially with an increase in load. At a load of 80 units, it takes 100 cycles for failure. When the load is halved, it takes 10000 cycles for failure. The load for which the failure will happen in 5000 cycles is ________.
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Correct answer: (B) 46.02
Explanation
Write the cycles as $N = N_0 e^{-bL}$. From $N(80) = 100$ and $N(40) = 10000$ we get $e^{40b} = 100$, so $b = \frac{\ln 100}{40} \approx 0.1151$. For 5000 cycles, which is half of $N(40)$, we need $e^{-b(L - 40)} = \frac{1}{2}$, so $L = 40 + \frac{\ln 2}{b} = 40 + 6.02 = 46.02$.