© T Madas

Describing the Accuracy of Measurements Part 1 © T Madas

111 ≤ L ≤ 113 Minimum Length: Maximum Length: 111 mm 113 mm The length of a pencil is measured as 112 ± 1 mm. What is the maximum and the minimum possible length of the pencil according to this information? Lower Bound Minimum Length: Maximum Length: 111 mm 113 mm Upper Bound 111 ≤ L ≤ 113 © T Madas

2.8 ≤ T ≤ 3.2 Minimum Thickness: Maximum Thickness: 2.8 mm 3.2 mm The thickness of a coin is measured as 3 ± 0.2 mm. What is the maximum and the minimum possible thickness of this coin according to this information? Lower Bound Minimum Thickness: Maximum Thickness: 2.8 mm 3.2 mm Upper Bound 2.8 ≤ T ≤ 3.2 © T Madas

2.985 ≤ T ≤ 3.015 Minimum Thickness: Maximum Thickness: 2.985 mm The thickness of an optical lens is measured as 3 ± 0.015 mm. What is the maximum and the minimum possible thickness of this lens according to this information? Lower Bound Minimum Thickness: Maximum Thickness: 2.985 mm 3.015 mm Upper Bound 2.985 ≤ T ≤ 3.015 © T Madas

Describing the Accuracy of Measurements Part 2 © T Madas

The weight of a chocolate bar and its packaging is 72 grams, correct to the nearest gram. What is the maximum and the minimum possible weight of this object according to this information? 68 69 70 71 72 73 74 75 Lower Bound Upper Bound grams Minimum Weight: Maximum Weight: 71.5 g 72.5 g 71.5 ≤ W < 72.5 © T Madas

The length of a pencil is 11.8 cm correct to 1 decimal place. What is the maximum and the minimum possible length of the pencil according to this information? 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 Lower Bound Upper Bound cm Minimum Length: Maximum Length: 11.75 cm 11.85 cm 11.75 ≤ L < 11.85 © T Madas

The weight of an empty can is 15.32 g correct to 2 decimal places What is the maximum and the minimum possible weight of the can according to this information? Repsi Cola 15.28 15.29 15.30 15.31 15.32 15.33 15.34 15.35 Lower Bound Upper Bound g Minimum Weight: Maximum Weight: 15.315 g 15.325 g 15.315 ≤ W < 15.325 © T Madas

The shutter time for a camera in cloudy conditions is 0 The shutter time for a camera in cloudy conditions is 0.017 seconds, correct to 3 decimal places. What is the maximum and the minimum possible shutter times according to this information? 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0.020 Lower Bound Upper Bound sec Minimum Weight: Maximum Weight: 0.0165 s 0.0175 s 0.0165 ≤ t < 0.0175 © T Madas

A specialised AC generator used in medicine is said to provide a constant voltage of 0.0475 volts, correct to 4 decimal places. What is the maximum and the minimum possible voltage that this instrument provides, according to this information? ESC LOAD ON ON BATT BYPASS FAULT generator 0.0472 0.0473 0.0474 0.0475 0.0476 0.0477 0.0478 0.0479 Lower Bound Upper Bound volts Minimum voltage: Maximum voltage: 0.04745 V 0.04755 V 0.04745 ≤ V < 0.04755 © T Madas

The length of a ladybird is 7 mm correct to the nearest mm What is the maximum and the minimum possible length of the ladybird according to this information? Lower Bound Upper Bound 10 mm 5 Minimum Length: Maximum Length: 6.5 mm 7.5 mm 6.5 ≤ L < 7.5 © T Madas

The weight of a male gorilla is measured as 285 kg, correct to the nearest 5 kg. What is the maximum and the minimum possible weight of this gorilla according to this information? 255 260 265 270 275 280 285 290 295 300 305 310 Lower Bound Upper Bound Kg Minimum Weight: Maximum Weight: 282.5 kg 287.5 kg 282.5 ≤ W < 287.5 © T Madas

The winning time of a 100 metre sprinter is 10 The winning time of a 100 metre sprinter is 10.2 seconds correct to the nearest tenth of a second. What is the fastest and the slowest possible winning times for this particular sprint? 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 Lower Bound Upper Bound sec Fastest Time: Slowest Time: 10.15 sec 10.25 sec 10.15 ≤ t < 10.25 © T Madas

The time exposure of a photographic film is measured as 0 The time exposure of a photographic film is measured as 0.036 seconds, correct to the nearest two thousandths of a second. What is the fastest and the slowest possible exposure times for this particular photographic film? 0.028 0.030 0.032 0.034 0.036 0.038 0.040 0.042 Lower Bound Upper Bound sec Fastest Time: Slowest Time: 0.035 sec 0.037 sec 0.035 ≤ t < 0.037 © T Madas

The weight of a bean is measured as 8 grams, correct to the nearest quarter of a gram. What is the maximum and the minimum possible weights for this bean? 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 Lower Bound Upper Bound grams Maximum Weight: Minimum Weight: 8.125 g 7.875 g 7.875 ≤ w < 8.125 © T Madas

Describing the Accuracy of Measurements Part 3 © T Madas

55 000 000 ≤ P ≤ 56 499 999 Minimum Population: Maximum Population: The population of the U.K. is estimated to be 56 000 000, correct to 2 significant figures. What is the maximum and minimum possible U.K. populations according to this information? Lower Bound Minimum Population: Maximum Population: 55 500 000 56 499 999 Upper Bound 55 000 000 ≤ P ≤ 56 499 999 © T Madas

5625 ≤ s < 5635 Minimum Speed: Maximum Speed: 5625 5635 The speed of a rocket is measured as to be 5630 km per hour, correct to 3 significant figures. What is the maximum and minimum possible rocket speeds according to this information? Lower Bound Minimum Speed: Maximum Speed: 5625 5635 Upper Bound 5625 ≤ s < 5635 © T Madas

14.5 ≤ P < 15.5 Minimum Length: Maximum Length: 14.5 cm 15.5 cm The length of a pens is measured as 15 cm, correct to the nearest whole number. What is the maximum and minimum possible lengths of this pen according to this information? Lower Bound Minimum Length: Maximum Length: 14.5 cm 15.5 cm Upper Bound 14.5 ≤ P < 15.5 © T Madas

Example © T Madas

Min Perimeter Max Perimeter Min Area Max Area 31.5 + 31.5 + 44.5 A rectangle measures 45 cm by 32 cm. These dimensions are correct to the nearest cm. Calculate the minimum possible perimeter of this rectangle Calculate the maximum possible perimeter of this rectangle Calculate the minimum possible area of this rectangle Calculate the maximum possible area of this rectangle Min Perimeter – 32.5 31.5 + 31.5 + 44.5 + 44.5 = 152 cm 31.5 Max Perimeter 44.5 – 45.5 32.5 + 32.5 + 45.5 + 45.5 = 156 cm Min Area 31.5 x 44.5 = 1401.75 cm2 Max Area 32.5 x 45.5 = 1478.75 cm2 © T Madas

Example © T Madas

x Minimum Length Maximum Length A right angled triangle has perpendicular sides equal to 140 mm and 84 mm. These dimensions are correct to 2 significant figures. Calculate the minimum and maximum possible lengths for the hypotenuse, correct to 1 d.p. Minimum Length x 2 = (83.5)2 + (135)2 c x 2 = 6972.25 + 18225 c x 2 = 25197.25 c – 145 x x = 158.7 mm 135 Maximum Length x 2 = (84.5)2 + (145)2 c x 2 = 7140.25 + 21025 c 83.5 – 84.5 x 2 = 28165.25 c x = 167.8 mm © T Madas

Example © T Madas

The formula is used to calculate the tension T exerted by an elastic spring. The following data is given, correct to 3 significant figures: The natural length of the spring L, is 0.386 m The extension of the spring x, is 0.156 m The modulus of elasticity λ, 1.52 N Find the maximum and the minimum possible values for the tension exerted by this spring, to 3 significant figures. 0.3855 0.1555 1.515 ≤ L ≤ ≤ x ≤ ≤ λ ≤ 0.3865 0.1565 1.525 To maximise T 1.525 x 0.1565 T = = 0.619 0.3855 © T Madas

To maximise T To minimise T The formula is used to calculate the tension T exerted by an elastic spring. The following data is given, correct to 3 significant figures: The natural length of the spring L, is 0.386 m The extension of the spring x, is 0.156 m The modulus of elasticity λ, 1.52 N Find the maximum and the minimum possible values for the tension exerted by this spring, to 3 significant figures. 0.3855 0.1555 1.515 ≤ L ≤ ≤ x ≤ ≤ λ ≤ 0.3865 0.1565 1.525 To maximise T 1.525 x 0.1565 T = = 0.619 0.3855 To minimise T 1.515 x 0.1555 T = = 0.610 0.3865 © T Madas

Approximation & bounds Quick Practice on Approximation & bounds © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 175 m nearest m 174.5 m 175.5 m 45 g nearest whole number 44.5 g 45.5 g 18 mm to the nearest mm 17.5 mm 18.5 mm 290 kg nearest ten 285 kg 295 kg 3300 s nearest one hundred 3250 s 3350 s 2.3 g to 1 d.p. 2.25 g 2.35 g 4.08 m to 2 d.p. 4.075 m 4.085 m 1.234 s to 3 d.p. 1.2335 s 1.2345 s 10.0 kg to 1 d.p. 9.95 kg 10.05 kg 0.003 g to 3 d.p. 0.0025 g 0.0035 g © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 34 cm nearest cm 33.5 cm 34.5 cm 56 mm nearest whole number 55.5 mm 56.5 mm 76 m to the nearest metre 75.5 m 76.5 m 220 kg nearest ten 215 kg 225 kg 7800 km nearest one hundred 7750 km 7850 km 10.8 s to 1 d.p. 10.75 s 10.85 s 1.77 kg to 2 d.p. 1.765 kg 1.775 kg 0.456 s to 3 d.p. 0.4555 s 0.4565 s 305 oz to the nearest 5 oz 302.5 oz 307.5 oz 602 m to the nearest 2 metres 601 m 603 m © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 26 cm to 2 s.f. 25.5 cm 26.5 cm 870 mm to 2 s.f. 865 mm 875 mm 315 m to the nearest five metres 312.5 m 317.5 m 220 kg nearest ten 215 kg 225 kg 1380 km to 3 s.f. 1375 km 1385 km 7.6 s to 1 d.p. 7.55 s 7.65 s 1.17 kg to 3 s.f. 1.165 kg 1.175 kg 0.024 s to 2 s.f. 0.0235 s 0.0245 s 12.5 oz to the nearest ½ oz 12.25 oz 12.75 oz 1.25 m to the nearest ¼ metre 1.125 m 1.375 m © T Madas

Approximation & bounds Quick Test on Approximation & bounds © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 175 m nearest m 174.5 m 175.5 m 45 g nearest whole number 44.5 g 45.5 g 18 mm to the nearest mm 17.5 mm 18.5 mm 290 kg nearest ten 285 kg 295 kg 3300 s nearest one hundred 3250 s 3350 s 2.3 g to 1 d.p. 2.25 g 2.35 g 4.08 m to 2 d.p. 4.075 m 4.085 m 1.234 s to 3 d.p. 1.2335 s 1.2345 s 10.0 kg to 1 d.p. 9.95 kg 10.05 kg 0.003 g to 3 d.p. 0.0025 g 0.0035 g © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 34 cm nearest cm 33.5 cm 34.5 cm 56 mm nearest whole number 55.5 mm 56.5 mm 76 m to the nearest metre 75.5 m 76.5 m 220 kg nearest ten 215 kg 225 kg 7800 km nearest one hundred 7750 km 7850 km 10.8 s to 1 d.p. 10.75 s 10.85 s 1.77 kg to 2 d.p. 1.765 kg 1.775 kg 0.456 s to 3 d.p. 0.4555 s 0.4565 s 305 oz to the nearest 5 oz 302.5 oz 307.5 oz 602 m to the nearest 2 metres 601 m 603 m © T Madas

Write the lower & upper bounds for these measurements Accuracy Lower Bound Upper Bound 26 cm to 2 s.f. 25.5 cm 26.5 cm 870 mm to 2 s.f. 865 mm 875 mm 315 m to the nearest five metres 312.5 m 317.5 m 220 kg nearest ten 215 kg 225 kg 1380 km to 3 s.f. 1375 km 1385 km 7.6 s to 1 d.p. 7.55 s 7.65 s 1.17 kg to 3 s.f. 1.165 kg 1.175 kg 0.024 s to 2 s.f. 0.0235 s 0.0245 s 12.5 oz to the nearest ½ oz 12.25 oz 12.75 oz 1.25 m to the nearest ¼ metre 1.125 m 1.375 m © T Madas

© T Madas