Maximizing Distance for a Three-Wheeled Vehicle with Spare Tyre
The problem asks for the maximum distance a three-wheeled vehicle can travel when it has 4 new tyres available (3 mounted + 1 spare), and each tyre has a maximum usage limit of 90 km.
Tyre Rotation Strategy
To maximize the total distance, we need to utilize all 4 tyres effectively. The vehicle uses 3 tyres at any given time. The core idea is to rotate the tyres, replacing a worn tyre with a less worn one (including the spare) to ensure no tyre exceeds its 90 km limit while covering the maximum possible ground.
Distance Calculation
We can think about the total "tyre-kilometres" available and required.
- Total number of tyres = 3 (on vehicle) + 1 (spare) = 4 tyres.
- Maximum distance per tyre = $90 \text{ km}$.
- Total available tyre-kilometres = $4 \text{ tyres} \times 90 \text{ km/tyre} = 360 \text{ km}$.
- Number of tyres in contact with the road at any time = 3.
- Let $D$ be the maximum total distance the vehicle can travel.
- Total tyre-kilometres consumed for distance $D$ = $3 \times D$.
- To achieve the maximum distance, the total tyre-kilometres consumed must equal the total available tyre-kilometres:
$3 \times D = 4 \times 90 \text{ km}$
$3D = 360 \text{ km}$
$D = \frac{360 \text{ km}}{3}$
$D = 120 \text{ km}$
Conclusion
By rotating the 4 available tyres, the three-wheeled vehicle can cover a maximum distance of 120 km.
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