Solving Volume Problems. ESSENTIAL QUESTION How do you find the volume of a figure made of cubes and prisms?
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1 L E S S O N 9.5 Solving Volume Problems ESSENTIAL QUESTION How do you find the volume of a figure made of cubes and prisms? Volume of a Triangular Prism The formula for the volume of a rectangular prism can be used for any prism. Volume of a Prism The volume V of a prism is the area of its base B times its height h. V = Bh Solve real-world and mathematical problems involving volume of three-dimensional objects composed of cubes, and right prisms. EXAMPLE 1 Bradley s tent is in the shape of a triangular prism. How many cubic feet of space are in his tent? Find the base area B of the triangular prism. Area of a triangle with base length b and height h Image Credits: PinkBlue/ Shutterstock Find the volume of the prism. The volume of Bradley s tent is 108 ft 3. Substitute 6 for b and 4 for h. Volume of a prism with base area B and height h Substitute 12 for B and 9 for h. 1. Analyze Relationships For a prism that is not a rectangular prism, how do you determine which sides are the bases? 2. Find the volume of the prism. Lesson
2 Volume of a Trapezoidal Prism Prisms are named for the polygons that form their bases. In this lesson, you will focus on prisms whose bases are either triangles or quadrilaterals other than squares and rectangles. EXAMPLE 2 Cherise is setting up her tent. Her tent is in the shape of a trapezoidal prism. How many cubic feet of space are in her tent? Find the base area B of the trapezoidal prism. Without calculating the volumes, how can you know whether Bradley s or Cherise s tent has a greater volume? Find the volume of the prism. Area of a trapezoid with bases of lengths b 1 and b 2 and height h Substitute 6 for b 1, 4 for b 2, and 4 for h. Volume of a prism with base area B and height h Substitute 20 for B and 9 for h. The volume of Cherise s tent is 180 ft Look for a Pattern How could you double the volume of the tent by doubling just one of its dimensions? 4. What If? How would doubling all the dimensions of the prism affect the volume of the tent? 5. Find the volume of the prism. 290 Unit 4
3 Volume of a Composite Solid You can use the formula for the volume of a prism to find the volume of a composite figure that is made up of prisms. EXAMPLE 3 Allie has two aquariums connected by a small square prism. Find the volume of the double aquarium. My Notes Find the volume of each of the larger aquariums. Volume of a prism Substitute 3 4 = 12 for B and 3 for h. Find the volume of the connecting prism. Volume of a prism Substitute 1 1= 1 for B and 2 for h. STEP 3 Add the volumes of the three parts of the aquarium. V = = 74 ft 3 The volume of the aquarium is 74 ft What If? Find the volume of one of the large aquariums on either end using another pair of opposite sides as the bases. Do you still get the same volume? Explain. 7. The figure is composed of a rectangular prism and a triangular prism. Find the volume of the figure. Lesson
4 Guided Practice 1. Find the volume of the triangular prism. (Example 1) 2. Find the volume of the trapezoidal prism. (Example 2) 3. Find the volume of the composite figure. (Example 3) Volume of rectangular prism = Volume of triangular prism = Volume of composite figure = Find the volume of each figure. (Examples 2 and 3) 4. The figure shows a barn that Mr. Fowler is building for his farm. 5. The figure shows a container, in the shape of a trapezoidal prism, that Pete filled with sand. ESSENTIAL QUESTION CHECK-IN 6. How do you find the volume of a composite solid formed by two or more prisms? 292 Unit 4
5 9.5 Independent Practice 7. A trap for insects is in the shape of a triangular prism. The area of the base is 3.5 in 2 and the height of the prism is 5 in. What is the volume of this trap? 8. Arletta built a cardboard ramp for her little brothers toy cars. Identify the shape of the ramp. Then find its volume. 9. Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal? 10. A gift box is in the shape of a trapezoidal prism with base lengths of 7 inches and 5 inches and a height of 4 inches. The height of the gift box is 8 inches. What is the volume of the gift box? Image Credits: The Photolibrary Wales/Alamy Images 11. Explain the Error A student wrote this statement: A triangular prism has a height of 15 inches and a base area of 20 square inches. The volume of the prism is 300 square inches. Identify and correct the error. Find the volume of each figure. Round to the nearest hundredth if necessary Multi-Step Josie has 260 cubic centimeters of candle wax. She wants to make a hexagonal prism candle with a base area of 21 square centimeters and a height of 8 centimeters. She also wants to make a triangular prism candle with a height of 14 centimeters. Can the base area of the triangular prism candle be 7 square centimeters? Explain. Lesson
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Name Period Date GEOMETRY AND MEASUREMENT Student Pages for Packet 1: GEO1.1 Congruence Plot simple figures on coordinate graphs, and determine their lengths and areas. Make conjectures about perimeters
17.2 Surface Area of Prisms
h a b c h a b c Locker LESSON 17. Surface Area of Prisms and Cylinders Texas Math Standards The student is expected to: G.11.C Apply the formulas for the total and lateral surface area of three-dimensional