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Course 2 10-5 Changing Dimensions Warm Up Find the surface area of each figure to the nearest tenth. Use 3.14 for . 1. a cylinder with radius 5 ft and height 11 ft 2. a sphere with diameter 6 ft 3. a rectangular prism 9 ft by 14 ft by 6 ft 502.4 ft 2 113.0 ft 2 528 ft 2

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Course 2 10-5 Changing Dimensions EQ: How do I find the volume and surface area of similar three-dimensional figures? M7G3.a Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts; M7G3.b Understand the relationships among scale factors, length ratios, and area ratios between similar figures. Use scale factors, length ratios, and area ratios to determine side lengths and areas of similar geometric figures;

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Course 2 10-5 Changing Dimensions Recall that similar figures are proportional. The surface areas of similar three-dimensional figures are also proportional. To see this relationship, you can compare the areas of corresponding faces of similar rectangular prisms.

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Course 2 10-5 Changing Dimensions Area of front of smaller prism Area of front of larger prism 3 · 5 6 · 10 15 (3 · 2) · (5 · 2) (3 · 5) · (2 · 2) 15 · 2 2 Each dimension has a scale factor of 2. A scale factor is a number that every dimension of a figure is multiplied by to make a similar figure. Remember!

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Course 2 10-5 Changing Dimensions The area of the front face of the larger prism is 2 2 times the area of the front face of the smaller prism. This is true for all of the corresponding faces. Thus it is also true for the entire surface area of the prisms. SURFACE AREA OF SIMILAR FIGURES The surface area of a three-dimensional figure A is equal to the surface area of a similar figure B times the square of the scale factor of figure A. surface area of figure A surface area of figure B (scale factor) 2 =

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Course 2 10-5 Changing Dimensions The surface area of a box is 35 in 2. What is the surface area of a larger, similar box that is larger by a scale factor of 7? Additional Example 1A: Finding the Surface Area of a Similar Figure S = 35 · 7 2 Multiply by the square of the scale factor. S = 35 · 49Evaluate the power. S = 1,715 Multiply. The surface area of the larger box is 1,715 in 2.

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Course 2 10-5 Changing Dimensions The surface area of a box is 1,300 in 2. Find the surface area of a smaller, similar box that is smaller by a scale factor of. Additional Example 1B: Finding the Surface Area of a Similar Figure 1212 S = 1,300 · 1212 2 1414 S = 325 The surface area of the smaller box is 325 in 2. Multiply by the square of the scale factor. Evaluate the power. Multiply.

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Course 2 10-5 Changing Dimensions Check It Out: Example 1A S = 50 · 3 2 Multiply by the square of the scale factor. S = 50 · 9 Evaluate the power. S = 450 Multiply. The surface area of the larger box is 450 in 2. The surface area of a box is 50 in 2. What is the surface area of a larger, similar box that is larger by a scale factor of 3?

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Course 2 10-5 Changing Dimensions 1313 S = 1,800 · 1313 2 1919 S = 200 The surface area of the smaller box is 200 in 2. Multiply by the square of the scale factor. Evaluate the power. Multiply. Check It Out: Example 1B The surface area of a box is 1,800 in 2. Find the surface area of a smaller, similarly shaped box that has a scale factor of.

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Course 2 10-5 Changing Dimensions The volumes of similar three-dimensional figures are also related. 3 ft 2 ft 1 ft 2 ft 6 ft 4 ft Volume of smaller box Volume of larger box 2 · 3 · 1 6 4 · 6 · 2 (2 · 2) · (3 · 2) · (1 · 2) (2 · 3 · 1) · (2 · 2 · 2) 6 · 2 3 Each dimension has a scale factor of 2. The volume of the larger box is 2 3 times the volume of the smaller box.

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Course 2 10-5 Changing Dimensions VOLUME OF SIMILAR FIGURES The volume of three-dimensional figure A is equal to the volume of a similar figure B times the cube of the scale factor of figure A. volume of figure A volume of figure B = (scale factor) 3

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Course 2 10-5 Changing Dimensions The volume of a child’s swimming pool is 28 ft 3. What is the volume of a similar pool prism that is larger by a scale factor of 4? Additional Example 2: Finding Volume Using Similar Figures V = 28 · 4 3 Multiply by the cube of the scale factor. V = 28 · 64 Evaluate the power. V = 1,792 ft 3 Multiply. Estimate V ≈ 30 · 60 Round the measurements. = 1,800The answer is reasonable.

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Course 2 10-5 Changing Dimensions Check It Out: Example 2 The volume of a small hot tube is 48 ft 3. What is the volume of a similar hot tub that is larger by a scale factor of 2? V = 48 · 2 3 Use the volume of the smaller prism and the cube of the scale factor. V = 48 · 8 Evaluate the power. V = 384 ft 3 Multiply. Estimate V ≈ 50 · 8 Round the measurements. = 400The answer is reasonable.

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Course 2 10-5 Changing Dimensions The sink in Kevin’s workshop measures 16 in. by 15 in. by 6 in. Another sink with a similar shape is larger by a scale factor of 2. There are 231 in 3 in 1 gallon. Estimate how many more gallons the larger sink holds. Additional Example 3: Problem Solving Application

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Course 2 10-5 Changing Dimensions Additional Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. Compare the capacities of two similar sinks, and estimate how much more water the larger sink holds. List the important information: The smaller sink is 16 in. x 15 in. x 6 in. The larger sink is similar to the small sink by a scale factor of 2. 231 in 3 = 1 gal

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Course 2 10-5 Changing Dimensions Additional Example 3 Continued 2 Make a Plan You can write an equation that relates the volume of the large sink to the volume of the small sink. The convert cubic inches to gallons to compare the capacities of the sinks. Volume of large sink = Volume of small sink · (a scale factor) 3

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Course 2 10-5 Changing Dimensions Additional Example 3 Continued Solve 3 Volume of small sink = 16 x 15 x 6 = 1,440 in 3 Convert each volume into gallons: Volume of large sink = 1,440 x 2 3 = 11,520 in 3 1,440 in 3 x ≈ 6 gallons 1 gal 231 in 3 11,520 in 3 x ≈ 50 gallons 1 gal 231 in 3 Subtract the capacities: 50 gal – 6 gal = 44 gal The large sink holds about 44 gallons more than the small sink.

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Course 2 10-5 Changing Dimensions Look Back 4 Double the dimensions of the small sink and find the volume: 32 x 30 x 12 = 11,520 in 3. Subtract the volumes of the two sinks: 11,520 – 1,440 = 10,080 in 3. Convert this measurement to gallons: 10,080 x ≈ 44 gal. Additional Example 3 Continued 1 gal 231 in 3

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Course 2 10-5 Changing Dimensions The bath tub in Ravina’s house measures 46 in. by 36 in. by 24 in. Another bath tub with a similar shape is smaller by a scale factor of. There are 231 in 3 in 1 gallon. Estimate how many more gallons the larger bath tub holds. Check It Out: Example 3 1212

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Course 2 10-5 Changing Dimensions Check It Out: Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. Compare the capacities of two similar tubs, and estimate how much more water the larger tub holds. List the important information: The larger tub is 46 in. x 36 in. x 24 in. 231 in 3 = 1 gal The smaller tub is similar to the larger tub by a scale factor of. 1212

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Course 2 10-5 Changing Dimensions Check It Out: Example 3 Continued 2 Make a Plan You can write an equation that relates the volume of the small tub to the volume of the large tub. The convert cubic inches to gallons to compare the capacities of the tubs. Volume of small tub = Volume of large tub · (a scale factor) 3

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Course 2 10-5 Changing Dimensions Check It Out: Example 3 Continued Solve 3 Volume of large tub = 46 x 36 x 24 = 39,744 in 3 Convert each volume into gallons: Volume of small tub = 39,744 x 0.5 3 = 4,968 in 3 39,744 in 3 x ≈ 172 gallons 1 gal 231 in 3 4,968 in 3 x ≈ 22 gallons 1 gal 231 in 3 Subtract the capacities: 172 gal – 22 gal = 150 gal The large tub holds about 150 gallons more than the small tub.

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Course 2 10-5 Changing Dimensions Look Back 4 Half the dimensions of the large tub and find the volume: 23 x 18 x 12 = 4,968 in 3. Subtract the volumes of the two tubs: 39,744 – 4,968 = 34,776 in 3. Convert this measurement to gallons: 34,776 x ≈ 150 gal. Check It Out: Example 3 Continued 1 gal 231 in 3

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Course 2 10-5 Changing Dimensions Lesson Quiz: Part I Given the scale factor, find the surface area to the nearest tenths of the similar prism. 1. The scale factor of the larger of two similar triangular prisms is 8. The surface area of the smaller prism is 18 ft 2. 2. The scale factor of the smaller of two similar triangular prisms is. The surface area of the larger prism is 600 ft 2. 66.7 ft 2 1,152 ft 2 1313

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Course 2 10-5 Changing Dimensions Lesson Quiz: Part II Given the scale factor, find the volume of the similar prism. 3. The scale factor of the larger of two similar rectangular prisms is 3. The volume of the smaller prism is 12 cm 3. 4. A food storage container measures 6 in. by 10 in. by 2 in. A similar container is reduced by a scale factor of. Estimate how many more gallons the larger container holds. 324 cm 3 About 0.5 gal 1212

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