How far can you deliver cargo using N trucks?
You have a fleet of 50 trucks. Each of them is fully fueled and can cover 100 km. How far can you deliver a certain item with their help? What will be the formula if you have N trucks at your disposal?
Not all people understand what is the task about: geographically it is a place where there are no gas stations. The only place where you can find fuel is the fuel tanks of trucks. You cannot change a truck for a Toyota Prius hybrid passenger car, for example. However, you have an opportunity to leave a truck without fuel and a driver. Finally, the only thing important in this task is to deliver a valuable item as far as possible.
There will be enough fuel to deliver each of 50 trucks for 100 km, thus for 50*100 = 5000 km. Should we use 5000 km as the answer? No, provided that you have a method to teleport the fuel from one truck’s fuel tank to another. Recall that each truck is fully gassed up and you cannot add more fuel until it is discharged.
Begin with a simple step. Imagine that you have not 50 trucks, but only one. Load the truck, get into and go on the road. In 100 km, the trip will be over for you.
Let’s now suppose that you have two trucks. Load the first one and you cannot forget about any problems for the first 100 km. But what then? Will the second truck be able to help you? Nope. It is in 100 km from you. It would have to follow you and its fuel tank will be empty in the same 100 km.
Maybe the first truck should have taken the second one in tow? Once the first truck has no fuel, you can transfer an item to the second truck and move on. OK, you will get 100 km more.
Thus, how far within such a coupling scheme can the first truck go? Hardly 100 km. It will need to carry weight twice as much as usual. The laws of physics state that at best, it will travel only half the distance.
Let’s consider another theory. Suppose that two trucks start going simultaneously and independently. In 50 km, the fuel tanks of both trucks will be half empty, but you can fuel fill up one of the vehicles completely. Pour fuel from one tank to another. Leave the empty truck and keep going in the filled up one 100 km more. Thus, the summary distance will be 150 km. Unlike with towage, you have to theoretical limitation here, and such an approach can be easily used in practice.
Considering the option with three trucks, the towage is in doubt, but the idea of fuel transfusion still works fine. Dispatch three trucks simultaneously. Let them stop for a third of the way at about 33.33 km. Each tank has 2/3 of fuel left. Transfer fuel from one truck to the tanks of the other two so that they become full again. Then dispatch these two filed trucks. You currently know that the maximum distance for them will be 150 km. If we add 33.33 km to this distance, the summary distance will be approximately 183.33 km.
The regularity becomes clear. One truck can drive 100 km. The second truck allows you to increase the total path by 100/2 = 50 km. The third truck increases the total path by 100/3 km. The fourth truck adds 100/4 km. This way, here is the formula to count distance for N trucks: 100 * (1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … 1 / N).
In this situation, the fractional part is known as the harmonic sequence. The sum of the harmonic sequence members can be easily calculated. If N is 50, the amount of this progression is 4.499… Multiply it by 100 km, and you will see that you can deliver cargo for 449.9 km having 50 trucks at your disposal.
Increasing N, the total sum grows up. Having a sufficient number of trucks, you can deliver cargo anywhere you want. However, when you increase N, the distance grows up slowly, but the energy efficiency ratio becomes very low. The thousandth truck will add only 1/100 km to the total distance of cargo delivery. However, this truck will pollute the atmosphere with carbon dioxide emissions in the same way as all other cars. The millionth truck will increase the trip distance by just a few centimetres.
Thus, we can conclude that the mentioned above solution option has the right to exist. You may wonder if there is another solution, right? Actually, yes, in case you can carry fuel and the cargo is not very heavy.
The task specifies the trucks that are intended for the large and heavy cargo carriage. Suppose you have a GMC or Ford truck. The own weight of such a fully gassed up and equipped vehicle is about 2250 kg. It is designed to safely transport such a heavy load.
The truck’s fuel tank holds about 30 fuel gallons that is approximately equal to 120 litres.
The key question is whether the fuel weighs less than the truck? Actually, less. The matter is 200/5000 is 1/25 of fueled truck weight without cargo.
It would be foolish to tow or drive a truck weighing 2250 kg when you are only interested in 120 litres of fuel. Is not it better to carry fuel in the truck body with the cargo? Maybe you can find fuel tanks or remove fuel tanks from other trucks and use them. A truck can carry fuel equivalent to fully fueling of25 trucks if the cargo is not very heavy.
It means that one such truck can carry the half truck fleet fuel consisting of 50 trucks. It can go for 25*100 or 2500 km. However, it will hardly accomplish this as the cargo will shorten this distance. Nevertheless, let’s conclude that such an option will allow the truck to overcome 1500 km. It is more than three times the size of 450 km with the fuel transfusion option and requires only one truck and one driver.