PUERTA | GATE-CS-2016 (Conjunto 1) | Pregunta 10

En un proceso, el número de ciclos hasta la falla disminuye exponencialmente con un aumento en la carga. Con una carga de 80 unidades, se necesitan 100 ciclos para fallar. Cuando la carga se reduce a la mitad, se necesitan 10000 ciclos para fallar. La carga para la cual ocurrirá la falla en 5000 ciclos es ________.

(A) 40.00

(B) 46.02
(C) 60.01
(D) 92.02

Respuesta: (B)
Explicación:

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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