Errors

Errors

Measurements are subject to errors.



Length of swimming pool = 25 metres to nearest metre



24.5 m 25.5 m

Lower Bound Upper Bound







24 m 25 m 26 m





Maximum error

0.5 m

Weight of package 4.3 kg

to 1 decimal place









4.25 kg 4.35 kg

Lower Bound Upper Bound







4.2 kg 4.3 kg 4.4 kg





Maximum error

0.05

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



Lower Bound Upper Bound

245 miles 255 miles







250 miles



–5 +5

miles 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







1350C 1450C

Lower Bound Upper Bound







1300C 1400C 1500C





– 50C + 50C





Temperature = 1400  50 C

If temperature of furnace = 1400C to 3 significant figures







1395C 1405C

Lower Bound Upper Bound







1390C 1400C 1410C





– 5 C + 5 C









Temperature = 1400  5 C

Car Park



45 m

Best estimate of perimeter

24 m 83 m

= 45 + 24 + 83 + 56

+ 128 + 80 24 + 56

= 80 m A B 56 m

= 416 m





Best estimate of area 45 + 83 = 128 m



Area of A  8045 = 3600 m2



Area of B  8356 = 4648 m2

But how accurate are

Total Area = 3600 + 4648 = 8248 m2 these estimates?

Car Park Upper Bounds



45.5 m

Upper Bound of perimeter

24.5 m 83.5 m

= 45.5 + 24.5 + 83.5 + 56.5

+ 129 + 81 24.5 + 56.5

= 81 m A B 56.5 m

= 420 m





Upper Bound of area 45.5 + 83.5 = 129 m



Upper Bound of Area of A  8145.5 = 3685.5 m2



Upper Bound of Area of B  83.556.5 = 4717.75 m2



Upper Bound of Total Area = 3685.5 + 4717.75 = 8403.25 m2

Car Park Lower Bounds



44.5 m

Lower Bound of perimeter

23.5 m 82.5 m

= 44.5 + 23.5 + 82.5 + 55.5

+ 127 + 79 23.5 + 55.5

= 79 m A B 55.5 m

= 412 m





Lower Bound of area 44.5 + 82.5 = 127 m



Lower Bound of Area of A  7944.5 = 3515.5 m2



Lower Bound of Area of B  82.555.5 = 4578.75 m2



Lower Bound of Total Area = 3515.5 + 4578.75 = 8094.25 m2

Car Park



45 m

Perimeter

24 m 83 m

Best estimate = 416 m

Lower Bound = 412 m

56 m

Upper Bound = 420 m





Total Area



Best Estimate = 8248 m2



Lower Bound = 8094.25 m2 Upper Bound = 8403.25 m2





Perimeter = 420 m (to 2 sf) Total Area = 8200 m2 (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 V  1 πr 2h

3



Best Estimate V  1 π  3.52  5.2 h

3



= 66.7 cm3

r



Upper Bound V  1 π  3.552  5.25 = 69.3 cm3

3

Lower Bound V  1 π  3.452  5.15 = 64.2 cm3

3





Best Estimate of Volume = 67 cm3 (to 2 sf)

Surface Area of cone r = 3.5 cm, h = 5.2 cm (to 2 sf)



Surface Area S  πr r  l 









l  r 2  h2



Best Estimate l  3.52  5.22 = 6.2682 cm l

h

S  π  3.5  3.5  6.2682 = 107.41 cm2











r

Upper Bound l  3.552  5.252 = 6.3376 cm



S π 3.553.55 6.3376 = 110.27 cm2





Lower Bound l  3.452  5.152 = 6.1988 cm



S π 3.453.45 6.1988 = 104.58 cm2



Best Estimate of Surface Area = 110 cm2 (to 2 sf)