Through the Hill

Engineers want the length of a tunnel through a hill. They cannot measure it directly, but from a point P away from the hill they can measure the distance to each end and the angle between the two lines of sight.

Ends 250 m and 310 m from P, 48° apart: ${t}^{2} = {250}^{2}+{310}^{2}-2\cdot 250\cdot 310\cdot \cos\left(48^{\circ}\right)$. The cosine of 48° is not a table value, so the board keeps the exact expression until you round: t is about 234.27 m.

Round only at the end. Rounding cos 48° to 0.67 first gives about 233.99 m, almost 30 cm out, which matters when two crews dig from both ends.