Hose Route Through a Storage Yard
A fire appliance is parked at the south gate of a rectangular storage yard. The yard is 60 m east-west and 40 m north-south. The gate is on the south boundary, 10 m east of the south-west corner. The fire is at the north-east corner, 5 m south of the north boundary and 5 m west of the east boundary. An east-west fence crosses the yard at 20 m north of the south boundary. The only gap in the fence is 25 m east of the west boundary. A 10 m wide strip along the east wall, from 10 m to 25 m north of the south boundary, is stacked with pallets and cannot be crossed. The appliance cannot enter the yard. Crews will lay a 38 mm hose from the appliance to the fire. Department procedure requires 10% extra hose beyond the measured route for couplings and changes of direction. Hose is available in 30 m lengths.
1. Using coordinates with the south-west corner as (0,0), calculate the shortest viable route and hose length from the appliance at the gate to the fire. Explain why the direct line is blocked.
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The gate is at (10,0), the fence gap is at (25,20), and the fire is at (55,35). A direct line from (10,0) to (55,35) has length √(45² + 35²) = √3250 ≈ 57.0 m. At the fence line y = 20, the direct line is at x = 10 + (20/35) × 45 ≈ 35.7, which is not the gap at x = 25, so the fence blocks it. The viable route is gate to fence gap to fire. Leg 1: √(15² + 20²) = 25.0 m. Leg 2: √(30² + 15²) = √1125 ≈ 33.5 m. Total measured route = 25.0 + 33.5 = 58.5 m. On leg 2, at x = 50 the route is at y = 20 + (25/30) × 15 = 32.5 m, which is north of the pallet strip ending at y = 25, so the route clears the pallets.
2. How many 30 m hose lengths should be requested, allowing 10% extra? Is a 60 m line adequate?
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The measured route is 58.5 m. The 10% allowance is 5.85 m, so the required hose is 58.5 + 5.85 = 64.35 m, approximately 64.4 m. A 60 m line is therefore short by about 4.4 m and is not adequate even before any unplanned detour. Hose is supplied in 30 m lengths, so two lengths give 60 m, which is insufficient; three lengths give 90 m, which provides 25.6 m of spare. The request should be for three 30 m lengths, not two. If the crew wants to minimize unused hose, they could request 90 m and lay only the required amount, but they should not plan on a 60 m line being sufficient.
3. At what point does the chosen route pass x = 50, and why does that matter for the pallet strip? What would change if the strip were extended to 35 m north?
Read the worked answer
On the second leg from (25,20) to (55,35), x = 50 occurs at a fraction (50 - 25)/(55 - 25) = 25/30 = 0.833 of the way along the leg. The y-coordinate is 20 + 0.833 × 15 = 32.5 m. The original pallet strip occupies y = 10 to 25 m, so the route passes 7.5 m north of its northern edge and is clear. If the strip were extended to 35 m north, the point (50, 32.5) would fall inside the strip and the direct diagonal would be blocked. The crew would need to detour west or north around the extended strip, which would add length and require a new measurement before requesting hose. The original 90 m request might still cover a modest detour, but it would need to be recalculated rather than assumed.