Moon Ride (Problem of the Week)
Problem Statement
In this problem, we were given the information that the year is 2097 and there has been a moon base built on the moon. A astronaut wants to make an exploratory trip around the moon and the car she is going to take has a fuel tank that holds enough fuel to last a fifth of the way there. In addition to that, the car can carry a tank of fuel with it. The tank of fuel can be opened and used or to deposited, unopened, on the moon's surface. The question we were asked was what is the minimum consumption of fuel to use in order to devise a way of making a round trip? As many preliminary trips as desired may be made to leave fuel tanks at strategic spots where they could be picked up and used later.
In this problem, we were given the information that the year is 2097 and there has been a moon base built on the moon. A astronaut wants to make an exploratory trip around the moon and the car she is going to take has a fuel tank that holds enough fuel to last a fifth of the way there. In addition to that, the car can carry a tank of fuel with it. The tank of fuel can be opened and used or to deposited, unopened, on the moon's surface. The question we were asked was what is the minimum consumption of fuel to use in order to devise a way of making a round trip? As many preliminary trips as desired may be made to leave fuel tanks at strategic spots where they could be picked up and used later.
Process and Solution
In the beginning, I decided to draw a circle and plot out five different points(A-E) to show the places should would maybe have to stop and refuel the car. After that I started mapping out different trips and where the fuel containers would be dropped off. After brainstorming, I figured out that the preliminary trips were going to be the most crucial part of the problem.
Preliminary Points
Part I: The car starts at point A (the base) with a tank full of fuel and a container of fuel, it travels to the halfway point in between point A and Point B to drop off the container of fuel and goes back to the base. This process is repeated three times, resulting in three containers at the halfway point. At the base the car is reloaded, the astronaut drives the car three quarters of the way in between point A and B. The astronaut turns back and drives the car back to the halfway point, arriving just out of gas. But because there are three tanks there, the astronaut will be able to reload the car with fuel with one container and be able to carry a container using another. The car will be driven to point B, dropping off the container there, the car will turn around and run out of gas at the halfway point in between point A and point B. Luckily, at the halfway point the third container of fuel will still be there and the car will be able to drive back to the base. This entire process will be repeated twice in the direction towards point B. (All of this was done to drop off two containers on point B.)
Part II: After completing that, the astronaut will have to repeat this process 6 times in the complete opposite direction towards point E. After completing this, the astronaut will drop off a container of fuel in between point A and point E, for future purposes. The astronaut has 6 containers of fuel on point E, starting from the base with a fully loaded car tank and a container of fuel in the car, the astronaut drives the car to point E and arrives out of fuel in the car. She refills the car with the container the car was carrying and picks up one of the six containers, loads it in the car, drops it off halfway in between point E and point D, and drives back to point E. At point E the car is loaded with the second container of the six containers and the third container of the six containers is used to fuel the car. The astronaut drives the car again to the halfway point in between point E and D and drops off a container of fuel it was carrying. The astronaut will drive back to point E to refuel the car with the fourth container of the six and load the fifth container onto the car. From there it will drop off the container it is carrying halfway in between point E and A and drive back to point E on the fuel remaining in the car. Yet again at point E, the car will be refueled by the sixth of the six containers. This time the car will be driven three quarters of the way from point E to point D. On the remaining fuel that is left, the car will drive to the halfway point in between point E and point D. At the halfway point with the three containers, the car will be refueled with the first container of the three and the second container of the three will be loaded onto the car. The newly refueled car will drive from the halfway point, in between point E and point D, to point D. At point D, the container the car was carrying will be dropped off and the car will ride back to the halfway point in between point E and D to pick up the last third container left there, to refuel the car. From there the car will drive to point E and start the complete round trip of the moon exploratory trip with a half tank of gas. (All of this was done to drop off a container of gas on point D.)
After the Preliminary Points there is one container on point D, one container on the halfway mark in between point E and point A, and two containers on point B.
Round-Trip
The car is beginning the round trip with a half tank of gas, but luckily we remembered to leave a new container of gas on the halfway point in between point A and point E. The car loads up the tank with the container and continues until it reaches point A (the base.) At the base, the astronaut refuels the half-empty tank. The car drives on to point B, arriving just out of fuel. There are two containers that were left on point B. The car refuels with one container and the astronaut loads the second container onto the car. The astronaut drives on and runs out of fuel when reaching point C. At point C, the car is refueled with the container it carried with it from point B. The car drives on and lasts until point D. There is one container on point D, the astronaut refuels the car and drives on till it reaches point E. When finally reaching point E, the astronaut has made the roundtrip of the moon!
IN TOTAL: 55.5 Gas Tanks Were Used
Preliminary Points
Part I: The car starts at point A (the base) with a tank full of fuel and a container of fuel, it travels to the halfway point in between point A and Point B to drop off the container of fuel and goes back to the base. This process is repeated three times, resulting in three containers at the halfway point. At the base the car is reloaded, the astronaut drives the car three quarters of the way in between point A and B. The astronaut turns back and drives the car back to the halfway point, arriving just out of gas. But because there are three tanks there, the astronaut will be able to reload the car with fuel with one container and be able to carry a container using another. The car will be driven to point B, dropping off the container there, the car will turn around and run out of gas at the halfway point in between point A and point B. Luckily, at the halfway point the third container of fuel will still be there and the car will be able to drive back to the base. This entire process will be repeated twice in the direction towards point B. (All of this was done to drop off two containers on point B.)
Part II: After completing that, the astronaut will have to repeat this process 6 times in the complete opposite direction towards point E. After completing this, the astronaut will drop off a container of fuel in between point A and point E, for future purposes. The astronaut has 6 containers of fuel on point E, starting from the base with a fully loaded car tank and a container of fuel in the car, the astronaut drives the car to point E and arrives out of fuel in the car. She refills the car with the container the car was carrying and picks up one of the six containers, loads it in the car, drops it off halfway in between point E and point D, and drives back to point E. At point E the car is loaded with the second container of the six containers and the third container of the six containers is used to fuel the car. The astronaut drives the car again to the halfway point in between point E and D and drops off a container of fuel it was carrying. The astronaut will drive back to point E to refuel the car with the fourth container of the six and load the fifth container onto the car. From there it will drop off the container it is carrying halfway in between point E and A and drive back to point E on the fuel remaining in the car. Yet again at point E, the car will be refueled by the sixth of the six containers. This time the car will be driven three quarters of the way from point E to point D. On the remaining fuel that is left, the car will drive to the halfway point in between point E and point D. At the halfway point with the three containers, the car will be refueled with the first container of the three and the second container of the three will be loaded onto the car. The newly refueled car will drive from the halfway point, in between point E and point D, to point D. At point D, the container the car was carrying will be dropped off and the car will ride back to the halfway point in between point E and D to pick up the last third container left there, to refuel the car. From there the car will drive to point E and start the complete round trip of the moon exploratory trip with a half tank of gas. (All of this was done to drop off a container of gas on point D.)
After the Preliminary Points there is one container on point D, one container on the halfway mark in between point E and point A, and two containers on point B.
Round-Trip
The car is beginning the round trip with a half tank of gas, but luckily we remembered to leave a new container of gas on the halfway point in between point A and point E. The car loads up the tank with the container and continues until it reaches point A (the base.) At the base, the astronaut refuels the half-empty tank. The car drives on to point B, arriving just out of fuel. There are two containers that were left on point B. The car refuels with one container and the astronaut loads the second container onto the car. The astronaut drives on and runs out of fuel when reaching point C. At point C, the car is refueled with the container it carried with it from point B. The car drives on and lasts until point D. There is one container on point D, the astronaut refuels the car and drives on till it reaches point E. When finally reaching point E, the astronaut has made the roundtrip of the moon!
IN TOTAL: 55.5 Gas Tanks Were Used
Extensions and Further Exploration
After working on this problem I was reminded a little bit of number line problems that I used to do. What made me think of it was how constantly in the problem I was moving back and forth between different points on the circle and how when using a number line you're moving left and right on the number line when showing addition and subtraction on the number line. Of course, this problem is definitely a longer and more drawn out version than using a number line, but I'm curious. Would it be easier to imagine the problem on a number line, rather than a circle diagram that I made? Perhaps if the problem was imagined on a number line, it would be easier to lay out the problem more clearly and the path that you are creating for the car would be more identifiable, because there is a specific starting point and ending point on a number line. However, if I had to redo the problem again I would still visualize it on a circle diagram, because when it is on the circle you can identify the multiple path options for the car to travel. If say I had made a number line diagram for the problem, I wouldn't have been able to see the path starting on point E. Maybe I wouldn't have gotten the answer at all if I put it on a number line format.
Reflection
Moon Ride (A Challenge POW) was a very challenging problem to solve. If not the hardest POW this year for me. The most important aspect of solving this POW was visualization. If I hadn't taken apart the problem and analyzed in a smaller way when making it into a circle graph, I never would've been able to make sense of the problem. The circle graph helped me visualize the problem in a different way. Starting small was also a big part of solving this problem, there were different steps that I split the problem up into. When I had accomplished the smaller steps of the problem, the bigger steps were a lot easier to understand and move forward on to solve the entire problem on.
Pictures of the Process are Down Below...