Errors Measurements are subject to errors Length of

Errors Measurements are subject to errors. Length of swimming pool = 25 metres to nearest metre 25. 5 m Upper Bound 24. 5 m Lower Bound 24 m 25 m Maximum error 0. 5 m 26 m

Weight of package 4. 25 kg Lower Bound 4. 2 kg 4. 3 kg to 1 decimal place 4. 35 kg Upper Bound 4. 3 kg Maximum error 0. 05 4. 4 kg

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 Accuracy Maximum Error Nearest 100 50 Nearest 10 5 Nearest whole number 0. 5 To 1 decimal place (nearest 0. 1) 0. 05 To 2 decimal places (nearest 0. 01) 0. 005

Length of journey = 250 miles to nearest 10 miles Upper Bound 255 miles Lower Bound 245 miles 250 miles – 5 miles +5 miles Length of journey = 250 ± 5 miles

Winning time in a race 36. 32 seconds to nearest 0. 01 second Maximum error = 0. 005 seconds Upper Bound = 36. 32 + 0. 005 = 36. 325 seconds Lower Bound = 36. 32 - 0. 005 = 36. 315 seconds Winning time = 36. 32 ± 0. 005 seconds

Temperature of furnace = 1400°C to 2 significant figures 1450°C Upper Bound 1350°C Lower Bound 1300°C 1400°C – 50°C + 50°C Temperature = 1400 ± 50 °C 1500°C

If temperature of furnace = 1400°C to 3 significant figures 1395°C Lower Bound 1405°C Upper Bound 1400°C 1390°C – 5°C + 5° C Temperature = 1400 ± 5 °C 1410°C

Car Park 45 m Best estimate of perimeter = 45 + 24 + 83 + 56 + 128 + 80 = 416 m 24 m 83 m 24 + 56 = 80 m A B 56 m 45 + 83 = 128 m Best estimate of area Area of A = 3600 m 2 Area of B = 4648 m 2 Total Area = 3600 + 4648 = 8248 m 2 But how accurate are these estimates?

Car Park Upper Bounds Upper Bound of perimeter = 45. 5 + 24. 5 + 83. 5 + 56. 5 24. 5 + 56. 5 + 129 + 81 = 81 m = 420 m Upper Bound of area 45. 5 m 24. 5 m A 83. 5 m B 45. 5 + 83. 5 = 129 m Upper Bound of Area of A = 3685. 5 m 2 Upper Bound of Area of B = 4717. 75 m 2 Upper Bound of Total Area = 3685. 5 + 4717. 75 = 8403. 25 m 2 56. 5 m

Car Park Lower Bounds Lower Bound of perimeter = 44. 5 + 23. 5 + 82. 5 + 55. 5 23. 5 + 55. 5 + 127 + 79 = 79 m = 412 m Lower Bound of area 44. 5 m 23. 5 m A 82. 5 m B 44. 5 + 82. 5 = 127 m Lower Bound of Area of A = 3515. 5 m 2 Lower Bound of Area of B = 4578. 75 m 2 Lower Bound of Total Area = 3515. 5 + 4578. 75 = 8094. 25 m 2 55. 5 m

Car Park Perimeter Best estimate = 416 m 45 m 24 m 83 m Lower Bound = 412 m 56 m Upper Bound = 420 m Total Area Best Estimate = 8248 m 2 Lower Bound = 8094. 25 m 2 Perimeter = 420 m (to 2 sf) Upper Bound = 8403. 25 m 2 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 h = 66. 7 cm 3 r Upper Bound = 69. 3 cm 3 Lower Bound = 64. 2 cm 3 Best Estimate of Volume = 67 cm 3 (to 2 sf)

Surface Area of cone r = 3. 5 cm, h = 5. 2 cm (to 2 sf) Surface Area Best Estimate = 6. 2682 cm l h = 107. 41 cm 2 Upper Bound r = 6. 3376 cm = 110. 27 cm 2 Lower Bound = 6. 1988 cm = 104. 58 cm 2 Best Estimate of Surface Area = 110 cm 2 (to 2 sf)