City Guilds Model Questions Vol1 2023 Engineering Question 22

A surveyor measures the height of a building using two different methods. In method A a theodolite measures the distance \(d=150.0\,\text{m}\) with an uncertainty of \(\pm0.1\,\text{m}\) and the vertical angle \(\theta=30.0^{\circ}\) with an uncertainty of \(\pm0.02^{\circ}\). The height is computed as \(h=d\tan\theta\). In method B a laser range‑finder gives the distance \(d'=150.2\,\text{m}\) with an uncertainty of \(\pm0.05\,\text{m}\) and an inclinometer measures the angle \(\theta'=30.5^{\circ}\) with an uncertainty of \(\pm0.05^{\circ}\). Using linear error propagation, determine which method yields the smaller maximum possible absolute error in the calculated height and state that error (in metres, rounded to three decimal places).