Problem #2
An ant is at the outside corner of a rectangular box (prism). If it wants to walk the shortest distance to the diagonally opposite corner, what path should it take?
The ant has a couple options for walking to the opposite corner of the box. To solve this problem I explored each path.
I represented the floor corners with A,B, C, and D. The ceiling corners will be represented by E, F, G, and H, with E being directly above A, F being
directly above B, Gbeing directly above C, and H being directly above D. (Draw a picture, it helps) the ant needs to go from the floor corner A to the ceiling corner G (if I’m not mistaken). The shortest path would be a diagonal line from A to G.
The ant has a couple options for walking to the opposite corner of the box. To solve this problem I explored each path.
I represented the floor corners with A,B, C, and D. The ceiling corners will be represented by E, F, G, and H, with E being directly above A, F being
directly above B, Gbeing directly above C, and H being directly above D. (Draw a picture, it helps) the ant needs to go from the floor corner A to the ceiling corner G (if I’m not mistaken). The shortest path would be a diagonal line from A to G.
Open Ended Problem
I. Problem Statement
There is a river boat that is supposed to carry 5 people and 50 pounds of supplies across the river. However, the weight that the boat can carry is only 300lbs at a time. Our job is to figure out how to carry across the people and supplies in the least amount of trips.
Key:
Supplies: 50 lbs
Al: 270 lbs
Zoe: 120 lbs
Chris: 180 lbs
Bob: 230 lbs
II. Process Description
When I first began solving this problem, it seemed fairly simple. I figured that it would take three trips total; Bob & Supplies, Chris & Zoe, and lastly, Al could go on his own. However, at this point, I did not realize that the boat will not travel unless there is someone in it to navigate. This meant that after each trip across, someone would have to go back to the take off point in order to pick up the next load. In order to continue solving the problem from here, I created a diagram along with the other students at my table and we came up with a logical solution.
III. Solution
Chris & Zoe -------> Zoe
<------------------------ Chris
Bob & Supplies ---> Bob & Supplies
<-------------------------- Zoe
Chris & Zoe ---------> Zoe
<-------------------------- Chris
Al -----------------------> Al
<-------------------------- Zoe
Chris & Zoe ---------> Chris & Zoe
The least amount of trips that can be taken is 9.
IIII. Self-Assessment and Reflection
I was extremely pleased with the way that my group members and I worked together on this open ended problem. When we first began, we didn't know that there must always be someone on the boat in order for it to take another trip, so we thought that we had solved the problem with simple addition techniques. When we realized that it was more complicated, we didn't want to rework and reconsider the new possibilities. However, after a small pep talk with ourselves, we got cracking again and solved it faster than we would have imagined. I made many contributions to the solving of this problem and am pleased with my perserverence.
There is a river boat that is supposed to carry 5 people and 50 pounds of supplies across the river. However, the weight that the boat can carry is only 300lbs at a time. Our job is to figure out how to carry across the people and supplies in the least amount of trips.
Key:
Supplies: 50 lbs
Al: 270 lbs
Zoe: 120 lbs
Chris: 180 lbs
Bob: 230 lbs
II. Process Description
When I first began solving this problem, it seemed fairly simple. I figured that it would take three trips total; Bob & Supplies, Chris & Zoe, and lastly, Al could go on his own. However, at this point, I did not realize that the boat will not travel unless there is someone in it to navigate. This meant that after each trip across, someone would have to go back to the take off point in order to pick up the next load. In order to continue solving the problem from here, I created a diagram along with the other students at my table and we came up with a logical solution.
III. Solution
Chris & Zoe -------> Zoe
<------------------------ Chris
Bob & Supplies ---> Bob & Supplies
<-------------------------- Zoe
Chris & Zoe ---------> Zoe
<-------------------------- Chris
Al -----------------------> Al
<-------------------------- Zoe
Chris & Zoe ---------> Chris & Zoe
The least amount of trips that can be taken is 9.
IIII. Self-Assessment and Reflection
I was extremely pleased with the way that my group members and I worked together on this open ended problem. When we first began, we didn't know that there must always be someone on the boat in order for it to take another trip, so we thought that we had solved the problem with simple addition techniques. When we realized that it was more complicated, we didn't want to rework and reconsider the new possibilities. However, after a small pep talk with ourselves, we got cracking again and solved it faster than we would have imagined. I made many contributions to the solving of this problem and am pleased with my perserverence.
The flight around the world
PROBLEM STATEMENT
There is a small island located on the top of the world. On this island there is a group of airplanes, each plane has enough fuel to take it halfway around the world. In order to make one plane go futher than halfway around the world, you are allowed to transfer fuel from one tank to another, granted the donor plane is able to make it safely back to the island. The only fuel source is located on the island. What is the smallest number of planes that can be used to ensure the flight of one plane around the world?
SOLUTION
The minimum amount of planes that can be used to ensure the flight of one plane around the world is three. Let’s call them planes A, B, and C. Once these three planes reach the ⅛ mark around the world, plane C refuels both plane B and plane A with ¼ tank of fuel each.
This means that plane A and B now have full tanks of gas, and plane C has exactly ¼ tank of gas which is the perfect amount of fuel it needs to take it back to the island.
Plane A and B can now continue their flight around the world (in a perfect circle).
Reaching ¼ around, plane B transfers another ¼ tank of fuel to plane A. This means that plane B now has a ½ tank of fuel which is the necessary amount to return it safely to the island.
Plane A is now at ¼ full and can reach ¾ with that amount of fuel.
Plane C, which returned to the island after the first fuel transfer, spends that time refueling their plane on the island. When plane A reaches the ¾ marker around the world, plance C must take off of the island in the opposite direction this time, meeting plane A at the ¾ marker.
When they meet, plane C must transfer a ¼ tank of fuel to plane A, leaving plane C with ¼ tank of fuel and plane A with ¼ tank of fuel which takes them to the ⅞ mark around the world.
Plane B, which had returned to the island, now refuels with ¾ tank, meets plane A and C at the ⅞ marker and transfers ¼ tank of fuel to both planes.
Plane A, B and C now each have ¼ tank of fuel and can make it back to the island safely.
This is definitely the minimum amount of planes that can be used. In order to properly refuel and save fuel, the two returning planes must only fuel up a little bit, not all of the way. Rather than having separate planes take off than the returning planes B and C, B and C are able to take the place of those extra planes and give plane A the necessary amount of fuel.
REFLECTION
At first I had great difficulty with this problem. I made it much more complicated than it needed be. After my group was handed the problem, my group members read it aloud and I interpretated the information they were reading. From this, I believed that in order to solve the problem, I would need to figure out how much fuel can stored in the average airplane, and how many miles per gallon it would get. I also thought that I needed to understand how many miles there were around the world when traveling in a perfect circle by flight.
Once my group members realized that I was overworking the problem, things became much easier and they helped lead me in the simple direction.
After fully understanding the proper problem, I did an excellent job of describing and articulating the situation. It was difficult keeping track of all of the planes and their fuel amounts, so I was also struggling to stay organized, but managed to. I also feel that I struggled with confidence at the beginning of this open ended problem which tested my patience, but in the end I was persistent and confident with my solution.
There is a small island located on the top of the world. On this island there is a group of airplanes, each plane has enough fuel to take it halfway around the world. In order to make one plane go futher than halfway around the world, you are allowed to transfer fuel from one tank to another, granted the donor plane is able to make it safely back to the island. The only fuel source is located on the island. What is the smallest number of planes that can be used to ensure the flight of one plane around the world?
SOLUTION
The minimum amount of planes that can be used to ensure the flight of one plane around the world is three. Let’s call them planes A, B, and C. Once these three planes reach the ⅛ mark around the world, plane C refuels both plane B and plane A with ¼ tank of fuel each.
This means that plane A and B now have full tanks of gas, and plane C has exactly ¼ tank of gas which is the perfect amount of fuel it needs to take it back to the island.
Plane A and B can now continue their flight around the world (in a perfect circle).
Reaching ¼ around, plane B transfers another ¼ tank of fuel to plane A. This means that plane B now has a ½ tank of fuel which is the necessary amount to return it safely to the island.
Plane A is now at ¼ full and can reach ¾ with that amount of fuel.
Plane C, which returned to the island after the first fuel transfer, spends that time refueling their plane on the island. When plane A reaches the ¾ marker around the world, plance C must take off of the island in the opposite direction this time, meeting plane A at the ¾ marker.
When they meet, plane C must transfer a ¼ tank of fuel to plane A, leaving plane C with ¼ tank of fuel and plane A with ¼ tank of fuel which takes them to the ⅞ mark around the world.
Plane B, which had returned to the island, now refuels with ¾ tank, meets plane A and C at the ⅞ marker and transfers ¼ tank of fuel to both planes.
Plane A, B and C now each have ¼ tank of fuel and can make it back to the island safely.
This is definitely the minimum amount of planes that can be used. In order to properly refuel and save fuel, the two returning planes must only fuel up a little bit, not all of the way. Rather than having separate planes take off than the returning planes B and C, B and C are able to take the place of those extra planes and give plane A the necessary amount of fuel.
REFLECTION
At first I had great difficulty with this problem. I made it much more complicated than it needed be. After my group was handed the problem, my group members read it aloud and I interpretated the information they were reading. From this, I believed that in order to solve the problem, I would need to figure out how much fuel can stored in the average airplane, and how many miles per gallon it would get. I also thought that I needed to understand how many miles there were around the world when traveling in a perfect circle by flight.
Once my group members realized that I was overworking the problem, things became much easier and they helped lead me in the simple direction.
After fully understanding the proper problem, I did an excellent job of describing and articulating the situation. It was difficult keeping track of all of the planes and their fuel amounts, so I was also struggling to stay organized, but managed to. I also feel that I struggled with confidence at the beginning of this open ended problem which tested my patience, but in the end I was persistent and confident with my solution.