Harbour Depth Windows for Two Vessels
A fishing harbour's channel depth is modelled by d(t) = 4 + 3 sin(πt/6) metres, where t is hours after midnight and 0 ≤ t ≤ 24. The harbour master has two rules. A trawler needs at least 5.5 m of water to enter; a ferry needs at least 6.25 m. The master claims: 'Both vessels can use the channel continuously for eight hours after the first safe entry, and the depth never falls below 1.5 m.' A trawler captain plans to enter at 02:00 and leave before the water drops below 5.5 m. A ferry operator wants the earliest time after midnight when the ferry can enter and the length of that first safe window. The port authority asks for exact expressions where possible and decimal checks to two places.
1. Find all intervals in 0 ≤ t ≤ 24 when the trawler can enter. Use them to test the master's eight-hour claim for the trawler, and find the minimum depth.
Read the worked answer
The trawler needs 4 + 3 sin(πt/6) ≥ 5.5, so sin(πt/6) ≥ 0.5.
Let θ = πt/6. In one period, sin θ ≥ 0.5 gives θ ∈ [π/6, 5π/6]. Thus πt/6 ∈ [π/6, 5π/6] + 2kπ, so t ∈ [1, 5] + 12k.
Within 0 ≤ t ≤ 24, the safe intervals are [1, 5] and [13, 17]. The first entry is at t = 1, but continuous safe water lasts only from 1 to 5, i.e. 4 hours, not 8. The master's eight-hour claim is false.
The minimum depth occurs when sin(πt/6) = -1, giving d = 4 - 3 = 1 m. This happens at t = 9 and t = 21. The claim that the depth never falls below 1.5 m is also false.
2. For the ferry, solve 4 + 3 sin(πt/6) ≥ 6.25. Give the earliest entry time and the length of that first window, then decide whether the master's 'eight hours' claim could be true for the ferry.
Read the worked answer
The ferry needs sin(πt/6) ≥ 0.75. Let α = arcsin(0.75) ≈ 0.8481 rad. In one period, θ ∈ [α, π - α], so
πt/6 ∈ [α, π - α] + 2kπ,
t ∈ [6α/π, 6(π - α)/π] + 12k.
Numerically, 6α/π ≈ 1.619 and 6(π - α)/π ≈ 4.381. The earliest entry is about 1.62 hours after midnight, i.e. about 01:37. The first safe window is [1.619, 4.381], whose length is about 4.381 - 1.619 = 2.76 hours.
This is far short of 8 hours, so the master's eight-hour claim is not true for the ferry either. The ferry has a shorter safe window than the trawler because its required depth is higher.