7. Define the terms (i) reliability; (ii) active redundancy; and (iii) standby redundancy. (4 marks) (b) Determine the reliability after 1 year of a system consisting of 3 identical pumps connected in parallel and 2 identical boilers also connected in parallel. For the system to operate normally, only 2 of the pumps and both boilers are required to work. All pumps and boilers fail randomly, and the pumps are operating in active redundancy mode. The MTBF for pumps and boilers are 8000 hours and 9500 hours, respectively. (8 marks) The general equation for active redundancy of parallel networks with m energized identical components of which n must work for system to operate at design conditions is: n n! ) R(t) = k! (12)(e~10)*(1 e-296a-4) k=m When a hospital that has five identical medical air compressor and requires three compressors to achieve peak demand, derive a reliability equation for this situation. (4 marks) (d) It is known that a manufacturing process produces 10% of sub-standard pipe fittings. If a sample of 30 fittings is taken at random from a box containing 1000 fittings, find the probability of the sample containing two or more sub-standard fittings. (4 marks) Chi-square Table Percentage points of the 2 distribution 0.995 0.99 0.9750.95 0.90 0.50 0.10 0.05 0.025 0.01 0.005 a Degrees of Freedom 1 2 3 4 5 0.71 1.06 6 2.09 0.000 0.0002 0.001 0.0039 0.016 0.45 2.71 3.84 5.02 6.63 7.88 0.010 0.020 0.051 0.103 0211 1.39 5.99 738 9.21 10.60 0.072 0.115 0216 0.3520584 237 6.25 781 935 1134 12.84 0.207 0.30 0.484 3.36 7.78 9.49 11.14 13.28 14.86 0.412 0.55 0.831 1.15 161 4.35 9.24 11.07 12.83 15.09 16.75 0.676 0.87 1.24 1.64 $.35 10.64 12.59 14.45 16.81 18.55 0.989 1.34 1.69 2.17 635 12.02 1407 16.01 18.48 20:28 1.34 1.65 2.18 2.73 3.49 734 13 36 1551 17.53 20.09 21.95 1.73 2.70 3.33 411 8.34 14.68 16.92 19.02 21.67 23.59 2.16 2.36 323 3.94 189 9,34 15.99 18.31 20.48 23.21 25.19 2.60 3.05 3.82 4.57 56 10.34 1728 19.68 21.92 24.72 26.76 3.07 3.57 4.30 5.23 6.30 11 34 18.55 21.03 2334 2622 28.30 3.57 4.11 5.01 5.89 704123119.81 2236 24.74 27.6929.82 4.07 4.66 5.63 6.57 7.79 13.34 21.06 23.68 26.12 29.14 31.32 4.60 5.23 6.26 7.26 8.35 14.34 22:31 25.00 27.49 30.58 32.80 5.14 5.81 6.91 7.96 9.31 1534 2254 26 30 28.85 32.00 34.27 5.70 6.41 7.56 8.67 10.09 16 34 24.77 2759 30.19 33.41 35.72 6.26 7.02 8.23 9.39 10.87 1734 2599 28.87 31.3 31.81 37.16 6.84 7.63 10.12 1165 18.34 27:20 30.14 32.85 36.19 38.58 743 9.59 10.RS 12.44 19:34 28.41 51.41 3417 37.57 40.00 8.03 890 1028 11.59 13.24 20.34 39.623267 35.48 38 93 41.40 8.64 9.54 10.98 12.34 14.04 21.34 30.81 33.92 36.78 40.29 42.80 9.26 10.20 11.69 13.09 14.85 2234 32.01 35.17 38.08 41.64 44.18 9.89 10.86 12.40 13.85 15.66 2334 33.20 36.42 39.36 43.98 45.56 10.52 11.52 13.12 14.61 16.47 24.34 34.38 37.65 40.65 44.31 46.93 11.16 12.20 13.84 15.38 1729 25.34 35.56 38.89 41.9245.6445 29 11.81 12.88 14.57 16.13 18.11 26.31 36.74 10.11 43.1946.96 19.64 12.46 13.37 15.31 16.93 1894 2734 37.92 41 34 44.46 48.28 $0.99 13.12 1426 1605 17.71 19.27 28 34 19.09 42.36 45.72 49.595234 13.19 14.95 16.29 18.49 20.60 29.34 40.26 43.7746.98 50.89 53.67 20.71 22.16 24.43 26.51 29.05 39 34 51.81 55.76 59 34 63.69 66.77 27.99 29.71 32.36 34.76 37.69 49.33 63.11 6750 71.41 76.15 79.49 35.53 37.48 43.19 36.86 $9.33 74.40 83.30 88,38 91.95 43.28 45.44 48.76 $1.74 5533 69.33 856390 53 95 02 100.43 1012 $1.17 53.54 57.15 60.39 64.28 19.33 96.58 101 88 106.63 112.33 116,3 59.20 61.75 65.65 69.13 73.39 89.33 107.57 113.15 118.14 124.12 128.3 67.33 70.06 74.22 77.99 2.36 99,33 118.50 124.34 129.56 135.81 1402 891 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 30 60 70 80 90 100 8.26 10.45 19.05 n