IOT POLY ENGINEERING I1-19 September 29, 2009 DRILL A: Two gears, A and B, are placed side by side. Gear A is rolled over gear B as shown. When gear A reaches the opposite side of gear B, which way will the orange gear tooth face? If gear A continues to roll around gear B until it returns to the original position, how many times will gear A have rotated around its own center? A B

DRILL A: Two Gears – SIMULATION SOLUTION IOT POLY ENGINEERING I1-19 DRILL A: Two Gears – SIMULATION SOLUTION When gear A reaches the opposite side of gear B, the orange gear tooth will be facing up, the same way it started. If gear A continues to roll around gear B until it returns to the original position, it will have rotated around its own center twice.

HOMEWORK PROBLEM #1 : SPIDER & FLY IOT POLY ENGINEERING I1-19 HOMEWORK PROBLEM #1 : SPIDER & FLY Given: A spider and a fly are in a room whose dimensions are 25 feet wide by 15 feet deep by 8 feet high. The spider is on the CEILING and the fly is on the FLOOR. If one corner of the room represents the origin (0,0,0) of an x-y-z coordinate system, then the spider is located at (20,8,-11 ) and the fly is located at (5,0,-7 ). See the given diagram. Problem: What is the MINIMUM DISTANCE that the spider must travel to reach the fly on the floor?

HOMEWORK PROBLEM #1 : SPIDER & FLY - SOLUTION IOT POLY ENGINEERING I1-19 HOMEWORK PROBLEM #1 : SPIDER & FLY - SOLUTION The spider is on the CEILING and the fly is on the FLOOR, but the spider is not directly above the fly. The shortest distance between the spider and the fly is a STRAIGHT LINE of about 17.5 feet, but the spider cannot travel that straight line because the spider cannot fly directly toward the fly on the floor. However, the spider can drop straight down to the floor by its spider silk. The distance to the floor is 8 feet. After the spider reaches the floor, its coordinates will be (20,0,-11). The following diagram shows both the spider and fly on the floor.

IOT POLY ENGINEERING I1-19 The Pythagorean Theorem can be used to find the distance between the spider and fly . SUMMARY: The spider would need to travel 8 feet DOWN from the ceiling and 15.5 feet ACROSS the floor for a TOTAL distance of 23.5 feet.

HOMEWORK PROBLEM #2 : SPIDER & FLY IOT POLY ENGINEERING I1-19 HOMEWORK PROBLEM #2 : SPIDER & FLY Given: A spider and a fly are in a room whose dimensions are 25 feet wide by 15 feet deep by 8 feet high. The spider is on the FLOOR and the fly is on the CEILING. If one corner of the room represents the origin (0,0,0) of an x-y-z coordinate system, then the spider is located at (5,0,-7) and the fly is located at (20,8,-11 ). See the given diagram. Problem: What is the MINIMUM DISTANCE that the spider must travel to reach the fly on the ceiling?

HOMEWORK PROBLEM #2 : SPIDER & FLY - SOLUTION IOT POLY ENGINEERING I1-19 HOMEWORK PROBLEM #2 : SPIDER & FLY - SOLUTION The spider is on the FLOOR and the fly is on the CEILING. The shortest distance between the spider and the fly is a STRAIGHT LINE of about 17.5 feet, but the spider cannot travel that straight line because the spider cannot fly directly toward the fly on the ceiling. The spider will have to travel across the floor, go up a wall, and then travel across the ceiling to the fly. 1. What path should the spider travel toward a wall? 2. Which wall should the spider climb? 3. What path should the spider travel up the wall? 4. What path should the spider travel across the ceiling?

HOMEWORK PROBLEM #2 : SPIDER & FLY - SOLUTION IOT POLY ENGINEERING I1-19 HOMEWORK PROBLEM #2 : SPIDER & FLY - SOLUTION 1. What path should the spider travel toward a wall? 2. Which wall should the spider climb? 3. What path should the spider travel up the wall? 4. What path should the spider travel across the ceiling? Traveling straight and perpendicular toward the closest wall seems logical, but it won’t give the shortest total path. The walls are all 8 feet high so it seems that no wall is better than another. Actually, one wall is better than the rest. Traveling straight up the wall seems logical, but it won’t give the shortest total path. 4. Once the spider reaches the ceiling, it should travel directly toward the fly. This seems logical, and is correct.

This problem can be solved by unfolding the room. IOT POLY ENGINEERING I1-19

At home, use a scissors to cut your paper as follows: IOT POLY ENGINEERING I1-19 At home, use a scissors to cut your paper as follows: TAPE Determine the location of the Fly on all 4 ceilings views.

This model will help visualize the problem. IOT POLY ENGINEERING I1-19 This model will help visualize the problem.

IOT POLY ENGINEERING I1-19 When the room is folded flat, we can see that the strategy of “directly to a wall, up the wall, and across the ceiling” does NOT give the shortest possible path! They aren’t straight.

IOT POLY ENGINEERING I1-19 Drawing straight lines from the spider’s location to the four fly locations gives shorter paths in all 4 cases, and one of these is the shortest possible path!

IOT POLY ENGINEERING I1-19 2.5 inches represents 25 feet.

September 29, 2009 HOMEWORK: Spider & Fly Problem IOT POLY ENGINEERING I1-19 HOMEWORK: Spider & Fly Problem Letter your answers. Use complete sentences. 1. What mathematical principles were involved? 2. What problem solving strategies were used? 3. What did you learn by solving this problem? 4. Describe in detail the path the spider must take from the floor to the ceiling.