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 - 2×250×310×cos(48°). 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.