Darren has 16 motorbikes with a tank that has a capacity to go 100 miles (when the tank is full).
→ All the motorbikes are initially fully fueled.
→ Each start from the same point.
→ Each bike has a rider on it.
Using these 16 motorbikes optimally, determine the maximum distance that Darren can travel.
Note:
It is not necessary for all the bikers to reach at that final point.
(assuming riders cannot disassemble the motorcycles and carry a gas tank with them; and assuming that Darren's wife is not Samantha, the witch from Bewitched)
I am submitting my entire process without editing except for spelling and cosmetic appearance:
1st plan: like a binary tree
16 bikes go 50 miles then 8 transfer fuel to the other 8
8 bikes go 50 miles then 4 transfer fuel to the other 4
4 bikes go 50 miles then 2transfer fuel to the other 2
2 bikes go 50 miles then 1 transfers fuel to the last bike
1 bike goes 100 miles.
Total distance: 300 miles
2nd plan: try a shorter distance before transferring fuel
16 bikes go 25 miles then 4 transfer fuel to the other 12
12 bikes go 25 miles then 3 transfer fuel to the other 9
9 bikes go 100/3 miles then 3 transfer fuel to the other 6
6 bikes go 100/3 miles then 2 transfer fuel to the other 4
4 bikes go 50 miles then 2 transfer fuel to the other 2
2 bikes go 50 miles then 1 transfers fuel to the last bike
1 bike goes 100 miles.
Total distance: 316 2/3 miles; better but surely we can do still better.
3rd plan: suspect optimal solution involves bikes quitting as soon as possible. The idea is that the average distance of all the bikes is constant, so if you want the last bike to go the farthest then many bikes must go not far at all.
16 bikes go 100/16 miles then one bike transfers fuel to the other 15
15 bikes go 100/15 miles then one bike transfers fuel to the others
14 bikes go 100/14 miles then one bike transfers fuel to the others
...
...
2 bikes go 100/2 miles
1 bike goes 100 miles
Total distance: 100 times the sum from i = 1 to 16 of 1/i
= 338.072899322899
I do not believe that I have proved that this solution is optimal, but I think it is.
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Posted by Larry
on 2022-12-27 09:17:51 |
Solution
