Errors Measurements are subject to errors. Length of swimming pool = 25 metres to nearest metre 25.5 m Upper Bound 25 m 26 m 24 m 24.5 m Lower Bound Maximum error 0.5 m 4.35 kg Upper Bound 4.3 kg 4.4 kg 4.2 kg 4.25 kg Lower Bound Maximum error 0.05 Weight of package 4.3 kg to 1 decimal place Errors in Measurements Lower Bound = lowest possible value Upper Bound = highest possible value When a measure is expressed to a given unit, the maximum error is half a unit. Errors in Measurements Nearest Accuracy Maximum Error Nearest 10 5 Nearest whole number 0.5 To 1 decimal place 0.05 (nearest 0.1) To 2 decimal places (nearest 0.01) Length of journey = 250 miles to nearest 10 miles Upper Bound 255 miles 250 miles Lower Bound 245 miles + 5 miles 5 miles Length of journey = 250 5 miles Winning time Maximum error = seconds Lower Bound = = seconds Upper Bound = = seconds Winning time in a race seconds to nearest 0.01 second = seconds Temperature of furnace = 1400 C to 2 significant figures Temperature = 1400 50 C 1450 C Upper Bound 1400 C 1500 C 1300 C 1350 C Lower Bound 50 C + 50 C If temperature of furnace = 1400 C to 3 significant figures Temperature = 1400 5 C 1405 C Upper Bound 1400 C 1410 C 1390 C 1395 C Lower Bound 5 C + 5 C 56 m 83 m 24 m 45 m Car Park Best estimate of perimeter = = 128 m = 80 m = 416 m Best estimate of area Area of A = 3600 m 2 A B Area of B = 4648 m 2 Total Area = = 8248 m 2 But how accurate are these estimates? 56.5 m 83.5 m 24.5 m 45.5 m Car Park Upper Bounds Upper Bound of perimeter = = 129 m = 81 m = 420 m Upper Bound of area Upper Bound of Area of A = m 2 A B Upper Bound of Area of B = m 2 Upper Bound of Total Area = = m 2 55.5 m 82.5 m 23.5 m 44.5 m Car Park Lower Bounds Lower Bound of perimeter = = 127 m = 79 m = 412 m Lower Bound of area Lower Bound of Area of A = m 2 A B Lower Bound of Area of B = m 2 Lower Bound of Total Area = = m 2 56 m 83 m 24 m 45 m Car Park Perimeter Total Area Best Estimate = 8248 m 2 Best estimate = 416 m Lower Bound = 412 m Upper Bound = 420 m Lower Bound = m 2 Upper Bound = m 2 Perimeter = 420 m (to 2 sf) Total Area = 8200 m 2 (to 2 sf) Example Find the volume and surface area of a cone with radius r = 3.5 cm, height h = 5.2 cm (to 1 dp) Volume Best Estimate = 66.7 cm 3 Upper Bound = 69.3 cm 3 Lower Bound = 64.2 cm 3 Best Estimate of Volume = 67 cm 3 (to 2 sf) h r Surface Area = cm 2 Upper Bound Best Estimate = cm = cm 2 = cm Surface Area of cone r = 3.5 cm, h = 5.2 cm (to 2 sf) Lower Bound = cm 2 = cm Best Estimate of Surface Area = 110 cm 2 (to 2 sf) h r l