Measurement errors - Answer key

Absolute error $= |3.35 - 3.2| = 0.15$ kg.
Relative error $= \dfrac{0.15}{3.2} = 0.046875$ .
Percentage error $= 0.046875 \times 100\% \approx 4.7\%$ .
Lower bound $= 15.75$ cm, upper bound $= 15.85$ cm.
Lower bound $= 23.5$ s, upper bound $= 24.5$ s.
Maximum possible error $= \dfrac{10}{2} = 5$ g.
Combined stated length $= 2.0$ m. Max error $= 0.05 + 0.05 = 0.10$ m. Lower bound $= 1.90$ m, upper bound $= 2.10$ m.
Absolute error $= |3.14 - 3.14159| = 0.00159$ . Percentage error $= \dfrac{0.00159}{3.14159} \times 100\% \approx 0.051\%$ .
Lower bound $= 21.5°$ C, upper bound $= 22.5°$ C.
Lower bound $= 45\,229.5$ km, upper bound $= 45\,230.5$ km.

Student A: $\dfrac{0.4}{12.0} \times 100\% = 3.33\%$ . Student B: $\dfrac{1.2}{46.0} \times 100\% = 2.61\%$ . Student B has the smaller percentage error.
Length bounds: $19.5$ to $20.5$ cm. Width bounds: $14.5$ to $15.5$ cm. Lower area $= 19.5 \times 14.5 = 282.75$ cm $^2$ . Upper area $= 20.5 \times 15.5 = 317.75$ cm $^2$ .
Speed $= \dfrac{400}{52.3} \approx 7.65$ m/s. Distance % error $= \dfrac{0.5}{400} \times 100\% = 0.125\%$ . Time % error $= \dfrac{0.05}{52.3} \times 100\% \approx 0.096\%$ . Max % error in speed $\approx 0.125\% + 0.096\% = 0.22\%$ .
Side bounds: $7.95$ cm to $8.05$ cm. Lower area $= 7.95^2 = 63.2025$ cm $^2$ . Upper area $= 8.05^2 = 64.8025$ cm $^2$ .
Percentage error scales the error to the size of the quantity, enabling fair comparison. A 1 cm error on a 10 cm measurement (10%) is far more significant than a 1 cm error on a 1000 cm measurement (0.1%), but the absolute errors are identical.
Volume $= \pi r^2 h = \pi \times 1.2^2 \times 3.0 \approx 13.57$ m $^3$ . Lower: $\pi \times 1.15^2 \times 2.95 \approx 12.26$ m $^3$ . Upper: $\pi \times 1.25^2 \times 3.05 \approx 14.97$ m $^3$ .
Bounds: $36.75°$ C to $37.25°$ C. Max error $= 0.25°$ C. Percentage error $= \dfrac{0.25}{37.0} \times 100\% \approx 0.68\%$ .

Distance percentage error $= \dfrac{|84.3 - 85.0|}{85.0} \times 100\% = \dfrac{0.7}{85.0} \times 100\% \approx 0.82\%$ . Area using measured distance $= 84.3^2 = 7106.49$ m $^2$ . Actual area $= 85.0^2 = 7225$ m $^2$ . Area percentage error $= \dfrac{|7106.49 - 7225|}{7225} \times 100\% \approx 1.64\%$ , which is approximately double the distance error (since area $\propto d^2$ ).
Density $= \dfrac{120}{50} = 2.4$ g/cm $^3$ . Mass % error $= \dfrac{0.5}{120} \times 100\% \approx 0.42\%$ . Volume % error $= \dfrac{1}{50} \times 100\% = 2.0\%$ . Max % error in density $\approx 0.42\% + 2.0\% = 2.42\%$ . Density $= 2.4 \pm 0.06$ g/cm $^3$ .
The error in $A - B$ is maximised when $A$ is at its upper bound and $B$ is at its lower bound (or vice versa). This gives the largest possible spread: $(A + \delta_A) - (B - \delta_B) = (A - B) + (\delta_A + \delta_B)$ . The individual errors add regardless of whether we add or subtract the measurements.
Max error $= 3 + 3 = 6$ m. For 200 m: % error $= \dfrac{6}{200} \times 100\% = 3\%$ . For 10 m: % error $= \dfrac{6}{10} \times 100\% = 60\%$ . The GPS is unreliable for short distances.

Radius $= 7.0$ cm, so area $= \pi \times 7.0^2 = 49\pi \approx 153.94$ cm $^2$ . Diameter % error $= \dfrac{0.1}{14.0} \times 100\% \approx 0.714\%$ . Since area $\propto d^2$ , area % error $\approx 2 \times 0.714\% = 1.43\%$ .
$g = \dfrac{2 \times 1.200}{0.495^2} = \dfrac{2.400}{0.245025} \approx 9.795$ m/s $^2$ . Distance % error $= \dfrac{0.005}{1.200} \times 100\% \approx 0.42\%$ . Time % error $= \dfrac{0.005}{0.495} \times 100\% \approx 1.01\%$ . Since $g \propto t^{-2}$ , time contributes $2 \times 1.01\% = 2.02\%$ . Total % error $\approx 0.42\% + 2.02\% = 2.44\%$ . This gives $g = 9.80 \pm 0.24$ m/s $^2$ . The accepted value $9.81$ falls within this range, so the result is acceptable.
Result $= 50.0 - 49.0 = 1.0$ . Max error $= 0.5 + 0.5 = 1.0$ . Percentage error $= \dfrac{1.0}{1.0} \times 100\% = 100\%$ . When two nearly equal quantities are subtracted, the result is small but the absolute error remains the sum of the original errors, leading to a very large relative error. This catastrophic cancellation should be avoided in practice by redesigning the measurement approach.
Map distance $= 4.2$ cm, so actual distance $= 4.2 \times 25\,000 = 105\,000$ cm $= 1050$ m. Lower bound: $(4.2 - 0.1) \times 25\,000 = 4.1 \times 25\,000 = 102\,500$ cm $= 1025$ m. Upper bound: $(4.2 + 0.1) \times 25\,000 = 4.3 \times 25\,000 = 107\,500$ cm $= 1075$ m. Actual distance is between 1025 m and 1075 m.

Tier 1