Mineral Economics, Mine Planning, Systems Engineering · Systems Engineering · 2 marks · Numerical answer
A pump has reliability $R(t)=\exp[-(t/1000)^{0.5}]$, where t is in years. If it comes with a six-month warranty, find the number of years for the pump to attain reliability 0.9, to two decimals.
Let the warranty endpoint be t₀=0.5 year. Conditional survival for an additional u years is R(t₀+u)/R(t₀)=0.9. Taking logarithms gives √((0.5+u)/1000)=√(0.5/1000)−ln0.9. Hence u=1000[√0.0005−ln0.9]²−0.5=15.81 years (total age 16.31 years). The original wording does not spell out conditioning; unconditional R(t)=0.9 instead gives 11.10 years and does not match its final key.
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