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Preview Warm Up California Standards Lesson Presentation

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**Warm Up Find the surface area of each rectangular prism.**

1. length 14 cm, width 7 cm, height 7 cm 2. length 30 in., width 6 in., height 21 in. 3. length 3 mm, width 6 mm, height 4 mm 4. length 37 in., width 9 in., height 18 in. 490 cm2 1872 in2 108 mm2 2322 in2

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California Standards MG2.3 Compute the length of the perimeter, the surface area of the faces, and the volume of a three-dimensional object built from rectangular solids. Understand that when the lengths of all dimensions are multiplied by a scale factor, the surface area is multiplied by the square of the scale factor and the volume is multiplied by the cube of the scale factor.

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Corresponding edge lengths of any two cubes are in proportion to each other because the cubes are similar. However, volumes and surface areas do not have the same scale factor as edge lengths. Each edge of the 2 ft cube is 2 times as long as each edge of the 1 ft cube. However, the cube’s volume, or capacity, is 23 = 8 times as large, and its surface area is 22 = 4 times as large as the 1 ft cube’s.

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**Additional Example 1A: Scaling Models That Are Cubes**

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the edge lengths of the two cubes 3 cm cube 1 cm cube 3 cm 1 cm Ratio of corresponding edges = 3 The length of the edges of the larger cube is 3 times the length of the edges of the smaller cube.

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**Additional Example 1B: Scaling Models That Are Cubes**

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the surface areas of the two cubes 3 cm cube 1 cm cube 54 cm2 6 cm2 Ratio of corresponding areas = 9 The surface area of the larger cube is 9 times that of the smaller cube.

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**Additional Example 1C: Scaling Models That Are Cubes**

A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the volumes of the two cubes 3 cm cube 1 cm cube 27 cm3 1 cm3 Ratio of corresponding volumes = 27 The volume of the larger cube is 27 times that of the smaller cube.

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Check It Out! Example 1A A 2 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the edge lengths of the large and small cubes 2 cm cube 1 cm cube 2 cm 1 cm Ratio of corresponding edges = 2 The length of the edges of the larger cube is 2 times the length of the edges of the smaller cube.

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Check It Out! Example 1B A 2 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the surface areas of the two cubes 2 cm cube 1 cm cube 24 cm2 6 cm2 Ratio of corresponding areas = 4 The surface area of the larger cube is 4 times that of the smaller cube.

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Check It Out! Example 1C A 2 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. the volumes of the two cubes 2 cm cube 1 cm cube 8 cm3 1 cm3 Ratio of corresponding volumes = 8 The volume of the larger cube is 8 times that of the smaller cube.

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**Additional Example 2A: Finding Surface Area and Volume of Similar Solids**

The surface area of a box is 1300 in2. What is the surface area of a similar box that is smaller by a scale factor of ? 1 2 1 2 Multiply by the square of the scale factor. S = 1300 · 2 1 4 = 1300 · Simplify the power. = 325 in2 Multiply.

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**Additional Example 2B: Finding Surface Area and Volume of Similar Solids**

The volume of a child’s swimming pool is ft3. What is the volume of a similar pool that is larger by a scale factor of 4? V = 28 · 43 Multiply by the cube of the scale factor. = 28 · 64 Simplify the power. = 1,792 ft3 Multiply.

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Check It Out! Example 2A The surface area of a box is 1800 in2. Find the surface area of a smaller, similarly shaped box that has a scale factor of ? 1 3 1 3 Multiply by the square of the scale factor. S = 1800 · 2 1 9 = 1,800 · Simplify the power. = 200 Multiply.

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Check It Out! Example 2B The volume of a small hot tub is 48 ft3. What is the volume of a similar hot tub that is larger by a scale factor of 2? V = 48 · 23 Use the volume of the smaller prism and the cube of the scale factor. = 48 · 8 Simplify the power. = 384 ft3 Multiply.

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**Additional Example 3: Business Application**

It takes 30 seconds for a pump to fill a cubic container whose edge measures 1 ft. How long does it take for the pump to fill a cubic container whose edge measures 2 ft? Find the volume of the 2 ft cubic container. V = 2 ft 2 ft 2 ft = 8 ft3 Set up a proportion and solve. 30 s 1 ft3 x 8 ft3 = Set up a proportion and solve. 30 8 = x Cross multiply. 240 = x Calculate the fill time. It takes 240 seconds, or 4 minutes, to fill the larger container.

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Check It Out! Example 3 It takes 30 seconds for a pump to fill a cubic container whose edge measures 1 ft. How long does it take for the pump to fill a cubic container whose edge measures 3 ft? Find the volume of the 3 ft cubic container. V = 3 ft 3 ft 3 ft = 27 ft3 Set up a proportion and solve. 30 s 1 ft3 x 27 ft3 = Set up a proportion and solve. 30 27 = x Cross multiply. 810 = x Calculate the fill time. It takes 810 seconds, or 13.5 minutes, to fill the larger container.

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Lesson Quiz: Part I A 10 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. 1. the edge lengths of the two cubes 2. the surface areas of the two cubes 3. the volumes of the two cubes 10:1 100:1 1000:1

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Lesson Quiz: Part II 4. The surface area of a prism is 600 ft2. What is the surface area of a similar prism that is smaller by a scale factor of ? 5. A cement truck is pouring cement for a new 4 in. thick driveway. The driveway is 90 ft long and 20 ft wide. How long will it take the truck to pour the cement if it releases 10 ft3 of cement per minute? 1 3 66.7 ft2 60 min

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