2. a. approximately cm 3 or 9p cm b. 20 layers c. approximately cm 3 or 180p cm Answers will vary.

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1 Answers Investigation ACE Assignment Choices Problem. Core Other Connections Problem. Core,, Other Applications 7, ; Connections 7 0; unassigned choices from previous problems Problem. Core 7 Other Connections, ; unassigned choices from previous problems Problem. Core, 0 Other Applications 9; Extensions ; unassigned choices from previous problems Adapted For suggestions about adapting Exercise and other ACE exercises, see the CMP Special Needs Handbook. Connecting to Prior Units : Covering and Surrounding; : Comparing and Scaling; 7: Variables and Patterns; 9 : Bits and Pieces II Applications NOTE: All answers were computed using. for p.. a. The cylinder made by taping the short sides together will have the greater volume (. cm vs. 0.7 cm ). One way to think about this is that the radius gets squared in finding volume, so we want to make this as big as possible (see the following note). However, at this stage students have not mastered any strategy for finding the volume of a cylinder. So they may focus incorrectly on height. b. If we imagine that the cylinders have tops and bottoms, then the cylinder made by taping together the short sides will have a larger top and bottom than the other cylinder. Since the area of the paper is the same for the two cylinders, the shorter cylinder has the greater surface area. For the Teacher Regardless of the size paper we begin with, the shorter cylinder will always have the greater surface area. A small amount of algebra shows this: If O is the length of our sheet of paper and w is the width, then the radius of our cylinder will be: O p. So the volume, after some computation, is: (O w) p. Putting the larger value in for O will maximize the volume. Your students will not produce this level of justification, but some may be curious if it is always true.. a. approximately. cm or 9p cm b. 0 layers c. approximately. cm or 0p cm. Answers will vary...p or 7.7. This is the surface area, in sq. cm, of Figure.. 9p or.. This is the area, in sq. cm, of Figure.. p. This is the volume for Figure. 7. a. approximately,00,000 cm b. approximately 7,000,000,000 cm (,000 times as much). a. height, radius, diameter, and circumference of the base b. area of the base and surface area c. volume 9... p 0,. cm Investigation Prisms and Cylinders 7

2 p.,0 cm. 00 =,00 in... For the cylinder in Exercise 9, surface area (.. p. p 0) 7. cm For the cylinder in Exercise 0, surface area (0 0 p + 0 p.),0. cm. a. No, because prisms with the same area of the base may have different dimensions for their bases. For example, an area of cm could be by or by for length and width. b. Yes, because the same area of the base means their radii are the same.. a. Possible sketch: b. approximately,9 ft c. approximately,9 ft. a. Possible sketch: cm 0 cm cm cm ft 0 ft cm 0 cm b. All such prisms, regardless of shape, will have a volume of 00 cm. c. Probably not. Many such prisms are possible because there are many polygons with an area of 0 cm. d. Yes, all correct prisms will have a volume of 00 cm. This is because we can think of the area of the base being one layer of unit cubes and the height being the number of identical layers required to fill the prism. Then the volume is the product of the area of the base and the height. These are both given in this problem.. a.. units. The area of the base is. sq. units. Multiply this by the height. b. 7 units. Each rectangle has an area of 0 sq. units. Each triangle has an area of. sq. units. So, 0 +. = a. approximately 9. cm or p cm b. approximately 9. cm using formula, or 70. using net for rectangular part. The prism has both greater volume and surface area. (Volume of prism and cylinder: cm,. cm. Surface area of prism and cylinder: 9 cm, 7. cm ) 9. a. Triangular prism: about 9.9 cubic inches,.7.. Square prism: about. cubic inches,.7.7. Pentagonal prism: 70. cubic inches,.. Hexagonal prism: about 7. cubic inches, ( ). b. These volumes are similar to those in Problem.. 0. Answers may vary. Possible answer: if height is feet, then the base area is 0 square feet and the radius is approximately.9 feet.. The area of the base needs to be 00 ft. One such base is 0 ft by 0 ft. 7 Filling and Wrapping

3 . a. If the cans are packed more efficiently (Figure ), more cans could fit in each layer. 0 cm 7 cm cm 0 cm Figure Figure b. Cylindrical box: volume is approximately,077. cm, surface area is approximately,0.0 cm. (Note: Being popcorn boxes, these do not have tops.) Rectangular box: volume is,0 cm. Surface area is,0 cm. c. Answers will vary. The cylindrical box will hold more popcorn. As a manager, you might choose the rectangular box because you will be going through less popcorn. However, its surface area is slightly more than the cylinder, which may increase the cost of the box, so the costs may offset each other. Connections. Jorge is correct. The area of this rectangle is p. (The point here is that p is a real number.) For the Teacher Students may struggle with this problem because they are unused to thinking of p as a length. Instead they, like Serge in the problem, think of p as a number useful in problems about circles. Nonetheless, p is a number that can theoretically designate a length just as easily as can.. The large size is the best buy. For $.00, you would get about 0.7 oz with the large size, about 0. oz with the medium size, or about 9. oz with the small size.. a. Answers will vary. b. Answers will vary. c. Measure to find the area of the base. Measure the height and multiply this by the area of the base.. If the cans are packed as though they were rectangular (Figure ),,00 cans would cover the floor in one layer. layers will fill the room for a total of,00 cans. 7. a. b. Circumference Circle Relationships Circumference Circle Relationships y x Investigation Prisms and Cylinders 7

4 c. d. Volume Cylinders With -cm Heights Volume (cm ) Cylinders With -cm Heights 0 y x e. Both graphs are increasing. As the diameter increases, the volume increases. However, the first graph is a straight line indicating that the rate of change is fixed from x to x +. In the second graph, the rate of change from one value of the diameter to the next in the table is not fixed. In fact, the volume grows faster in this graph as the diameter increases.. The container s volume should be between the volumes of the large and the small cylinders with radius. and, or between 90.9 and 9.. In fact, it should be around the middle value of 90.9 and 9., respectively, which is.9. Another way to find its volume is to find the difference between the cones with radius. and. The height of the small cone could be found by using the proportion of the container s difference of radii and height, which is. =. Then x =. The container s x volume is the difference of the two cones, which is =. cm. 9. a. = of the small containers to fill the larger. b. Division. Because the question asks how many one-thirds are in three and a third. 0. a. = of the small containers to fill the larger. b. Division. Because the question asks how many two-thirds are in two and two-fifths.. a. = containers b. of the volume of the tank c. a little more than 0 containers. a. = or containers b. of the volume of the tank c. 00 = 7 containers 9 Extensions. a. Radius: cm Height: 0 cm b. Approximately. cm, the difference between the volume of the box (0 cm ) and the volume of the can (. cm ). c. about 0.7 (Formally, it would be p to.) d. Specific answers will vary. However, for any such can and box, the ratio will be as in Question C. This is because the volume of the box is (r) h while the volume of the can is pr h. So, the ratio becomes pr h to r h, or p to.. a. The volume of this prism, as for any other prism, is found by multiplying the area of the base by the height. Here, this is 0 cm. b. Exact volume 7 Filling and Wrapping

5 . To make a net of a cylinder, we must know its radius and height. Because we know the height and volume of a cylinder, by the formula V = pr h, we can find the radius by dividing V by p h and then find its square root. Possible Answers to Mathematical Reflections. The volume of a rectangular prism can be found by multiplying the number of cubes that would fit in the bottom layer of the prism by the number of layers that would fit in the prism. Or, it can be found by multiplying the area of the base by the height of the prism or by multiplying length by width by height; V O w h.. a. The volume of a cylinder can be found by multiplying the area of its base by the height of the cylinder. Because the base is a circle, the area of the base is pr. The height of the cylinder can be represented by h. Thus, the volume of a cylinder can be found by using V = pr h. b. The surface area of a cylinder can be found by finding the sum of the area of the two circular ends, pr and then the area of the lateral surface and adding them together. The lateral surface is a rectangle, with its length the circumference of the circular base, pr, and the width of the rectangle is the height of the cylinder. Thus, the surface area of the cylinder can be represented by: SA = pr + prh.. For rectangular prisms and cylinders, the volume is the area of the base times the height. For prisms with bases that are polygons, finding the area depends on what kind of polygon it is. For cylinders, the base is always a circle for which we have a formula that gives us the area.. For surface area, you have to find the areas of all parts of a net that will cover the prism or cylinder. A cylinder always has a simple net made up of a square or rectangle and two circles. Prisms have more faces to consider identical bases plus the lateral faces. Investigation Prisms and Cylinders 77