How can you use formulas to find the volume of rectangular prisms?
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1 ? Name 9.2 Essential Question Formulas for Volume How can you use formulas to find the volume of rectangular prisms? Algebraic Reasoning 5.4.G Also 5.6.A, 5.6.B MATHEMATICAL PROCESSES 5.1.B, 5.1.D, 5.1.E Connect The base of a rectangular prism is a rectangle. You know that area is measured in square units, and that the area of a rectangle can be found by multiplying the length and the width. Volume is the measure of the amount of space a solid figure occupies. Volume is measured in cubic units, such as cubic feet or cu ft. When you build a prism and add each layer of cubes, you are adding a third dimension, height. The area of the base is _ sq units. Unlock the Problem Margo is modeling a building using 1-centimeter cubes. The model has a rectangular base and a height of the 5 cubes. What is the volume of the rectangular prism that Margo built? Use centimeter cubes to show Margo s model. Count the total number of cubes after each layer. Use the information to complete the table. Height (in layers) Volume (in cubic centimeters) 8 16 What multiplication pattern do the numbers in the table show? 8 Why do we multiply by 8? You can find the volume of a prism in cubic units by multiplying the number of square units in the base shape by the number of layers, or its height. Write a formula for finding the volume of a rectangular prism. Use B for the area of the base and h for the height. So, the volume of Margo s rectangular prism is _ cubic centimeters. Module 9 355
2 Relate Height to Volume You can use the formula for the area of a rectangle to rewrite the formula for the volume of a rectangular prism. What are the dimensions of the base of the box? What operation can you use to find the area of the base shape? Toni stacks cube-shaped beads that measure 1 centimeter on each edge in a storage box. The box can hold 6 layers of 24 cubes with no gaps or overlaps. What is the volume of Toni s storage box? One Way Use V = Bh. The volume of each bead is _ cubic cm. The storage box has a base with an area of _ square cm. The height of the storage box is _ centimeters. The volume of the storage box is ( _ _ ), or _ cubic cm. Base area Another Way Use length, width, and height. The base is a rectangle. Replace B with an expression for the area of the base shape. V = B h B = base area; l = length; w = width V = ( _ _ ) h Base area The base has a length of _ centimeters Math Talk Mathematical Processes Describe one way you can check if the volume you calculated using the formulas is correct. and a width of _ centimeters. The height is _ centimeters. The volume of the storage box is ( _ _ ) _, or _ _, or _ cubic cm. Base area So, the volume of Toni s storage box is _ cubic cm. 356
3 Name Share and Show Find the volume. 1. The length of the rectangular prism is. 2. The width is. So, the area of the base is. The height is. So, the volume of the prism is in. 4 in. 2 cm 6 in. 1 in. 2 in. Problem Solving 4. Connect What happens to the volume of a rectangular prism if you double the height? Give an example. 5. Write Math Explain how the two formulas for the volume of a rectangular prism V = Bh and V = l w h are related. Use a model with centimeter cubes to justify your reasoning. Problem Solving 6. Multi-Step Rich is building a travel crate for his dog, Thomas, a beagle-mix who is about 30 inches long, 12 inches wide, and 24 inches tall. For Thomas to travel safely, his crate needs to be a rectangular prism that is about 12 inches greater than his length and width, and 6 inches greater than his height. What is the volume of the travel crate that Rich should build? Module 9 Lesson 2 357
4 Daily Assessment Task Mathematical Processes Fill in the bubble completely to show your answer. 7. Sandra orders a box of cube-shaped beads. The beads are neatly stacked in layers, with the same number of beads in each layer. Each layer is 12 beads across and 7 beads wide. There are 4 layers. How many beads does she receive? A 23 beads C 76 beads B 336 beads D 88 beads 8. Which equation can you use to find the number of cubes in the rectangular prism? A V = C V = B V = D V = 2 (11 8 5) 9. Multi-Step How many cubic units of material were used to make the two rectangular prisms? A 144 cubic units C 8,640 cubic units B 60 cubic units D 204 cubic units TEXAS Test Prep What is the volume of the rectangular prism at the right? A 125 cu in. C 155 cu in. B 35 cu in. D 175 cu in. 7 in.
5 Homework and Practice Name Algebraic Reasoning 5.4.G Also 5.6.A, 5.6.B MATHEMATICAL PROCESSES 5.1.B, 5.1.D, 5.1.E 9.2 Formulas for Volume Find the volume cm 2 cm 1 in. 3 in. 4 in cm cm 4 in. 3 in. 3 in. Problem Solving 7. A small refrigerator fits into a cabinet that measures 2 feet wide, 2 feet deep, and 4 feet high. What is the volume of the cabinet? 8. Mr. Otis built a storage shed. The shed has a length of 5 meters, a width of 3 meters, and a height of 4 meters. His goal was for the shed to have a volume greater than 50 cubic meters. Did Mr. Otis meet his goal? Explain. Module 9 Lesson 2 359
6 Lesson Check TEXAS Test Prep Fill in the bubble completely to show your answer. 9. What is the volume of the rectangular prism shown below? 10. Hannah used centimeter cubes to build the model shown below. 8 cm Which equation can you use to find the volume of Hannah s model? A V = A B C D 48 cu cm 84 cu cm 288 cu cm 384 cu cm B V = (6 7) + 3 C V = (6 + 7) 3 D V = A shipping clerk packs a box of cube-shaped notepads. He packs 8 layers of notepads with 8 rows of 6 notepads in each layer. How many notepads does the clerk pack? A 336 C 96 B 112 D The number cubes the fifth-grade math classes use are packed into a box. When the box is full, it has 5 rows of 4 cubes with 6 layers of cubes. Mrs. Benson sees that one layer of cubes is missing from the box. How many number cubes are in the box? A 96 C 90 B 100 D Multi-Step Alexis uses toy blocks to build a model of a building. Each toy block is 1 cubic inch. The first three floors of the model are made up of 6 rows of 4 blocks. Floors four through eight are made up of 4 rows of 4 blocks. What is the volume of the model? 360 A 104 cu in. C 88 cu in. B 152 cu in. D 136 cu in. 14. Multi-Step Aaden is packing boxes into a carton that is 8 inches long, 8 inches wide, and 4 inches tall. The boxes are 2 inches long, 1 inch wide, and 1 inch tall. How many boxes will fit into the carton? A 128 C 64 B 256 D 512
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