1. 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

2. 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

3. 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.

4. Errors in Measurements Nearest 100 50 Accuracy Maximum Error Nearest 10 5 Nearest whole number 0.5 To 1 decimal place 0.05 (nearest 0.1) To 2 decimal places 0.005 (nearest 0.01)

5. 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

6. Winning time Maximum error = 0.005 seconds Lower Bound = 36.32 - 0.005= 36.315 seconds Upper Bound = 36.32 + 0.005= 36.325 seconds Winning time in a race 36.32 seconds to nearest 0.01 second = 36.32  0.005 seconds

7. 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

8. 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

9. 56 m 83 m 24 m 45 m Car Park Best estimate of perimeter = 45 + 24 + 83 + 56 + 128 + 80 45 + 83 = 128 m 24 + 56 = 80 m = 416 m Best estimate of area Area of A = 3600 m 2 A B Area of B = 4648 m 2 Total Area = 3600 + 4648 = 8248 m 2 But how accurate are these estimates?

10. 56.5 m 83.5 m 24.5 m 45.5 m Car Park Upper Bounds Upper Bound of perimeter = 45.5 + 24.5 + 83.5 + 56.5 + 129 + 81 45.5 + 83.5= 129 m 24.5 + 56.5 = 81 m = 420 m Upper Bound of area Upper Bound of Area of A = 3685.5 m 2 A B Upper Bound of Area of B = 4717.75 m 2 Upper Bound of Total Area = 3685.5 + 4717.75 = 8403.25 m 2

11. 55.5 m 82.5 m 23.5 m 44.5 m Car Park Lower Bounds Lower Bound of perimeter = 44.5 + 23.5 + 82.5 + 55.5 + 127 + 79 44.5 + 82.5 = 127 m 23.5 + 55.5 = 79 m = 412 m Lower Bound of area Lower Bound of Area of A = 3515.5 m 2 A B Lower Bound of Area of B = 4578.75 m 2 Lower Bound of Total Area = 3515.5 + 4578.75 = 8094.25 m 2

12. 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 = 8094.25 m 2 Upper Bound = 8403.25 m 2 Perimeter = 420 m (to 2 sf) Total Area = 8200 m 2 (to 2 sf)

13. 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

14. Surface Area = 107.41 cm 2 Upper Bound Best Estimate = 6.2682 cm = 110.27 cm 2 = 6.3376 cm Surface Area of cone r = 3.5 cm, h = 5.2 cm (to 2 sf) Lower Bound = 104.58 cm 2 = 6.1988 cm Best Estimate of Surface Area = 110 cm 2 (to 2 sf) h r l