Let s try drawing cross-sections of an everyday object, such as a coffee cup. Sketch the cross-sections at each of the indicated heights.

Transcription

1 Cross Sections Let s try drawing cross-sections of an everyday object, such as a coffee cup Sketch the cross-sections at each of the indicated heights Continued on back

2 QUESTIONS: A. Horizontal slices of a solid are shown at various levels arranged from highest to lowest. What could the solid be? B. One modern technology that requires a great understanding of cross sections and volume is the 3D printer. 3D printers perform a process called additive manufacturing, in which layers of plastic are stacked to form a precise 3-dimensional object. Explain the difference in the 3D printing process of the ring pictured in Figure 1 and the ring in Figure 2 if they are printed exactly as shown. Figure 1 Figure 2 C. Suppose you want to make a 3D printing of a cone. Does it matter if the vertex is placed at the top or at the bottom? Assume that the 3D printer places each new layer on top of the previous layer. Explain your response. D. The medium that a 3D printer will use to print (i.e., the ink ) is a type of plastic that comes in coils of tubing that have a diameter of 3mm. John wants to make 3D printings of a cone with radius 22cm and height 3cm. The length of the ink is 25m meters. About how many cones can John make? (HINT: Use the PARCC Reference sheet to convert all the measurements to centimeters)

3 Euler s Formula Let s make a roadmap! Here are some definitions you ll need to know. 1. Roadmap - a collection of line segments (possibly curved) in a plane, each with exactly two endpoints, such that line segments may only touch at their endpoints NOT an endpoint endpoint 2. Streets the line segments in your map (edges in Euler s Formula) 3. Corners the endpoints of your lines (vertices in Euler s Formula) 4. Blocks the regions into which the street divide the plane (faces in Euler s Formula) NOTE: the regions outside the roadmap is also a block (Ex. The road map above has 3 blocks) 5. Connected a roadmap is connected if given any two corners, you can trace a path from one corner to the other without ever leaving the streets of the roadmap. This roadmap is NOT connected because there is no path between corner A and B A B THEOREM: If a roadmap is connected, then Euler s Formula applies. QUESTIONS: A. If a connected roadmap has 10 blocks and 7 corners, how many streets are there? streets B. If a connected roadmap has 50 streets and 30 corners, how many blocks are formed? blocks C. Draw a connected roadmap with exactly 3 streets and 2 blocks. D. Can you form a connected roadmap with exactly five corners such that every pair of corners is connected by a street? Use Euler s formula to verify.

4 Cavilieri s Principle The management of an ocean life museum will choose to include either Aquarium A or Aquarium B in a new exhibit. Can you help them decide which to choose? Aquarium A is a right cylinder with a diameter of 10 feet and a height of 5 feet. Covering the lower base of Aquarium A is an underwater mountain in the shape of a 5-foot-tall right cone. This aquarium would be built into a pillar in the center of the exhibit room. Aquarium B is half of a 10-foot-diameter sphere. This aquarium would protrude from the ceiling of the exhibit room. A. How many cubic feet of water will Aquarium A hold? B. For each aquarium, what is the area of the water s surface when filled to a height of 3 ft? Aquarium A: Aquarium B: C. Explain the relationship between the volume of Aquarium A and the volume of Aquarium B using Cavilieri s Principle. D. Does it matter which aquarium the management decides to install? Explain.

5 Cross Sections Let s try drawing cross-sections of an everyday object, such as a coffee cup Sketch the cross-sections at each of the indicated heights Continued on back

6 QUESTIONS: E. Horizontal slices of a solid are shown at various levels arranged from highest to lowest. What could the solid be? Answers will vary. Could be a sphere with a hollow center. F. One modern technology that requires a great understanding of cross sections and volume is the 3D printer. 3D printers perform a process called additive manufacturing, in which layers of plastic are stacked to form a precise 3-dimensional object. Explain the difference in the 3D printing process of the ring pictured in Figure 1 and the ring in Figure 2 if they are printed exactly as shown. For the first ring, the cross-sections are circles or regions between concentric circles. For the second ring, the cross-sections are stretched circles and then two separted regions. Figure 1 Figure 2 G. Suppose you want to make a 3D printing of a cone. Does it matter if the vertex is placed at the top or at the bottom? Assume that the 3D printer places each new layer on top of the previous layer. Explain your response. If the vertex is at the top, new layers will always be supported by old layers. If the vertex is at the bottom, new layers will hang over previous layers. H. The medium that a 3D printer will use to print (i.e., the ink ) is a type of plastic that comes in coils of tubing that have a diameter of 3mm. John wants to make 3D printings of a cone with radius 22cm and height 3cm. The length of the ink is 25m meters. About how many cones can John make? (HINT: Use the PARCC Reference sheet to convert all the measurements to centimeters) 14 cones

7 Euler s Formula Let s make a roadmap! Here are some definitions you ll need to know. 6. Roadmap - a collection of line segments (possibly curved) in a plane, each with exactly two endpoints, such that line segments may only touch at their endpoints NOT an endpoint endpoint 7. Streets the line segments in your map (edges in Euler s Formula) 8. Corners the endpoints of your lines (vertices in Euler s Formula) 9. Blocks the regions into which the street divide the plane (faces in Euler s Formula) NOTE: the regions outside the roadmap is also a block (Ex. The road map above has 3 blocks) 10. Connected a roadmap is connected if given any two corners, you can trace a path from one corner to the other without ever leaving the streets of the roadmap. This roadmap is NOT connected because there is no path between corner A and B A B THEOREM: If a roadmap is connected, then Euler s Formula applies. QUESTIONS: E. If a connected roadmap has 10 blocks and 7 corners, how many streets are there? 15 streets F. If a connected roadmap has 50 streets and 30 corners, how many blocks are formed? 22 blocks G. Draw a connected roadmap with exactly 3 streets and 2 blocks. H. Can you form a connected roadmap with exactly five corners such that every pair of corners is connected by a street? Use Euler s formula to verify. 5 corners + 11 blocks does NOT equal 10 streets + 2

8 Cavilieri s Principle The management of an ocean life museum will choose to include either Aquarium A or Aquarium B in a new exhibit. Can you help them decide which to choose? Aquarium A is a right cylinder with a diameter of 10 feet and a height of 5 feet. Covering the lower base of Aquarium A is an underwater mountain in the shape of a 5-foot-tall right cone. This aquarium would be built into a pillar in the center of the exhibit room. Aquarium B is half of a 10-foot-diameter sphere. This aquarium would protrude from the ceiling of the exhibit room. E. How many cubic feet of water will Aquarium A hold? 252ft 3 F. For each aquarium, what is the area of the water s surface when filled to a height of 3 ft? Aquarium A: approximately 66 square feet (see next page for explanation) Aquarium B: approximately 66 square feet (see next page for explanation) G. Explain the relationship between the volume of Aquarium A and the volume of Aquarium B using Cavilieri s Principle. Cavalieri s principle states that two 3-dimensional solids of the same height are equal in volume if the areas of their horizontal cross-sections at every height h are equal. Since both aquariums stand 5 feet tall, and since the area of the water s surface when filled to a height of 3 feet is the same for each aquarium, the volumes of water must be equal when both aquariums are filled to full capacity. H. Does it matter which aquarium the management decides to install? Explain. Since both tanks hold the same amount of water, it is now a matter of preference. The management would have to decide how much money to spend and which aquarium was a better fit for the museum.

9 Part B Explanation: Cavilieri s Principle