Lesson 26: Volume of Composite Three-Dimensional Objects

Transcription

1 Lesson 26: Volume of Composite Three-Dimensional Objects Student Outcomes Students compute volumes of three-dimensional objects composed of right prisms by using the fact that volume is additive. Lesson Notes Lesson 26 is an extension of work done in the prior lessons on volume as well as an extension of work started in the final lesson of Module 3 (Lesson 26). Students have more exposure to composite figures such as prisms with prism shaped holes or prisms that have smaller prisms removed from their volumes. Furthermore, in applicable situations, students compare different methods to determine composite volume. This is necessary when the entire prism can be decomposed into multiple prisms or when the prism hole shares the height of the main prism. Classwork Example 1 (4 minutes) Example 1 Find the volume of the following three-dimensional object composed of two right rectangular prisms. Volume of object Volume of top prism Volume of bottom prism Volume of top prism: Volume of bottom prism: m 3 m 3 m 3 m 3 The volume of the object is m 3 m 3 There are different ways the volume of a composite figure may be calculated. If the figure is like the figure in Example 1, where the figure can be decomposed into separate prisms and it would be impossible for the prisms to share any one dimension, the individual volumes of the decomposed prisms can be determined and then summed. If, however, the figure is similar to the figure in Exercise 1, there are two possible strategies. In Exercise 1, the figure can be decomposed into two individual prisms, but a dimension is shared between the two prisms in this case the height. Instead of calculating the volume of each prism and then taking the sum, we can calculate the area of the entire base by decomposing it into shapes we know and then multiplying the area of the base by the height. Date: 4/9/14 277

2 Exercise 1 (4 minutes) Exercise 1 Find the volume of the following three-dimensional figure composed of two right rectangular prisms. Volume of object Volume of back prism Volume of front prism Volume of back prism: Volume of front prism: The volume of the object is Exercise 2 (10 minutes) Exercise 2 The right trapezoidal prism is composed of a right rectangular prism joined with a right triangular prism. Find the volume of the right trapezoidal prism shown in the diagram using two different strategies. Strategy #1 The volume of the trapezoidal prism is equal to the sum of the volumes of the rectangular and triangular prisms. Volume of object Volume of rectangular prism Volume of triangular prism Volume of rectangular prism: Volume of triangular prism: MP. 1 The volume of the object is Strategy #2 The volume of a right prism is equal to the area of its base time its height. The base consists of a rectangle and a triangle. cm cm 2 cm Volume of object Volume of object: cm cm cm 2 cm 2 cm 2 cm 2 The volume of the object is Date: 4/9/14 278

3 Write a numeric expression to represent the volume of the figure in Strategy 1. Write a numeric expression to represent the volume of the figure in Strategy 2. How do the numeric expressions represent the problem differently? The first expression is appropriate to use when individual volumes of the decomposed figure are being added together, whereas the second expression is used when the area of the base of the composite figure is found and then multiplied by the height to determine the volume. What property allows us to show that these representations are equivalent? The distributive property. Example 2 (10 minutes) Example 2 Find the volume of the right prism shown in the diagram whose base is the region between two right triangles. Use two different strategies. Strategy #1 The volume of the right prism is equal to the difference of the volumes of the two triangular prisms. Volume of object Volume large prism Volume small prism : : The volume of the object is. Strategy #2 The volume of a right prism is equal to the area of its base times its height. The base is the region between two right triangles. Volume of object Volume of object: The volume of the object is. Date: 4/9/14 279

4 Write a numeric expression to represent the volume of the figure in Strategy 1. Write a numeric expression to represent the volume of the figure in Strategy 2. How do the numeric expressions represent the problem differently? The first expression is appropriate to use when the volume of the smaller prism is being subtracted away from the volume of the larger prism, whereas the second expression is used when the area of the base of the composite figure is found and then multiplied by the height to determine the volume. What property allows us to show that these representations are equivalent? The distributive property. Example 3 (10 minutes) Example 3 A box with a length of ft., a width of ft., and a height of ft. contains fragile electronic equipment that is packed inside a larger box with three inches of styrofoam cushioning material on each side (above, below, left side, right side, front, and back). a. Give the dimensions of the larger box. Length ft., width ft., and height ft. b. Design styrofoam right rectangular prisms that could be placed around the box to provide the cushioning; i.e., give the dimensions and how many of each size are needed. Possible answer: Two pieces with dimensions ft. ft. in. and four pieces with dimensions ft. ft. in. c. Find the volume of the styrofoam cushioning material by adding the volumes of the right rectangular prisms in the previous question. ft 3 ft 3 ft 3 ft 3 ft 3 ft 3 d. Find the volume of the styrofoam cushioning material by computing the difference between the volume of the larger box and the volume of the smaller box. ft 3 ft 3 ft 3 Closing (2 minutes) To find the volume of a three-dimensional composite object, two or more distinct volumes must be added together (if they are joined together) or subtracted from each other (if one is a missing section of the other). There are two strategies to find the volume of a prism: Find the area of the base, then multiply times the prism s height; decompose the prism into two or more smaller prisms of the same height and add the volumes of those smaller prisms. Exit Ticket (5 minutes) Date: 4/9/14 280

5 Name Date Exit Ticket A triangular prism has a rectangular prism cut out of it from one base to the opposite base, as shown in the figure. Determine the volume of the figure, provided all dimensions are in millimeters. Is there any other way to determine the volume of the figure? If so, please explain. Date: 4/9/14 281

6 Exit Ticket Sample Solutions A triangular prism has a rectangular prism cut out of it from one base to the opposite base, as shown in the figure. Determine the volume of the figure, provided all dimensions are in millimeters. Is there any other way to determine the volume of the figure? If so, please explain. Possible response: Volume of the triangular prism: Volume of the rectangular prism: Volume of composite prism: The calculations above subtract the volume of the cut out prism from the volume of the main prism. Another strategy would be to find the area of the base of the figure, which is the area of the triangle less the area of the rectangle, and then multiply by the height to find the volume of the prism. Problem Set Sample Solutions 1. Find the volume of the three-dimensional object composed of right rectangular prisms. Volume object Volume top and bottom prisms Volume middle prism Volume of top and bottom prisms: Volume of middle prism: The volume of the object is 2. A smaller cube is stacked on top of a larger cube. An edge of the smaller cube measures cm in length, while the larger cube has an edge length three times as long. What is the total volume of the object? Volume of object Volume small cube Volume large cube The total volume of the object is. Date: 4/9/14 282

7 3. Two students are finding the volume of a prism with a rhombus base but are provided different information regarding the prism. One student receives Figure 1, while the other receives Figure 2. Figure 1 Figure 2 a. Find the expression that represents the volume in each case; show that the volumes are equal. Figure 1 Figure 2 3 mm b. How does each calculation differ in the context of how the prism is viewed? In Figure 1, the prism is treated as two triangular prisms joined together. The volume of each triangular prism is found and then doubled, whereas in Figure 2, the prism has a base in the shape of a rhombus, and the volume is found by calculating the area of the rhomboid base and then multiplying by the height. 4. Find the volume of wood needed to construct the following side table composed of right rectangular prisms. Volume of bottom legs: Volume of vertical legs: Volume of table top: The volume of the table is. 5. A plastic die (singular for dice) of a game has an edge length of cm. Each face of the cube has the number of cubic cut outs as its marker is supposed to indicate (i.e., the face marked has cut outs). What is the volume of the die? Number of cubic cut outs: Volume of cut out cubes: Volume of large cube: The total volume of the die is. Date: 4/9/14 283

8 6. A wooden cube with edge length inches has square holes (holes in the shape of right rectangular prisms) cut through the centers of each of the three sides as shown in the figure. Find the volume of the resulting solid if the square for the holes has an edge length of inch. Think of making the square holes between opposite sides by cutting three times: The first cut removes, and the second and third cuts each remove. The resulting solid has a volume of. 7. A right rectangular prism has each of its dimensions (length, width, and height) increased by. By what percent is its volume increased? The larger volume is of the smaller volume. The volume has increased by. 8. A solid is created by putting together right rectangular prisms. If each of the side lengths is increase by, by what percent is the volume increased? If each of the side lengths is increased by, then the volume of each right rectangular prism is multiplied by. Since this is true for each right rectangular prism, the volume of the larger solid,, can be found by multiplying the volume of the smaller solid,, by ; i.e.. This is an increase of. Date: 4/9/14 284