在一个过程中，失效的周期数随着负载的增加呈指数减少。在 80 个单位的负载下，故障需要 100 个周期。当负载减半时，需要 10000 个周期才会失效。在 5000 个周期内发生故障的负载是________。
(一) 40.00
(乙) 46.02
(C) 60.01
(四) 92.02答案：(乙)
解释：
For exponential dependence we must have functions of form
f(x) = a.(k power x) : a,k are constants and f(0)=a
Let us assume: C = cycles for failure ; L = Load; a, k = constants
according to question C is an exponential function of L
Hence we can say,
1) C = a x (k^L) ---- > in case C is increasing exponentially
2) C = a / (k^L) ---- > in case C is decreasing exponentially
we take 2nd equation and apply log to both side we get:
log C + L x (log k) = log a ...... by logarithmic property
given (C=100 for L=80) and (C=10000 for L=40)
apply these to values to above equation, this gives us 2 equations
and 2 variables:
log a = log 100 + (80 x log k)...............(1)
log a = log 10000 + (40 x log k)...........(2)
we have 2 variables and 2 equations hence we find
log a = 6
log k= 1/20
now find, [ L = (log a - log C) / (log k) ] for C=5000 cycles
=> L = (6 - log 5000)*20 = 46.0206 ~ 46.02 => [ANS]
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